# A Novel Linear Active Disturbance Rejection Control Design for Air-Breathing Supersonic Vehicle Attitude System with Prescribed Performance

Authors: Chao Ming; Xiaoming Wang
Journal: International Journal of Aerospace Engineering
 (2020)

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## 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.


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## Body

## 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 Section2. 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:
(1)ϑ̇=ωz,ω̇z=MzJz,α=ϑ−θ,where the pitch moment Mz is defined as
(2)Mz=q¯SLm0+mzαα+mzα2α2+q¯SLmzδzδz,where elevator deflection δz is the control input, and m0, mzα, mzα2, andmzδz are the aerodynamic coefficients.For simplicity, defining the statex1,x2=ϑ,ωz and combining with (1) and (2), the ASV attitude control system can be described with the strict feedback state-space form:
(3)ẋ1=x2,ẋ2=f+bu,where
(4)f=q¯SLm0+mzαα+mzα2α2Jz+Δb=q¯SLmzδzJzu=δzHere, the termf is described as “total disturbance,” which consists of the unknown dynamic uncertainty and external disturbance.Remark 1.
The uncertainty termΔ is mainly the result of parameter uncertainties and external disturbances, e.g., the aerodynamic parameters perturbations and wind interference.Assumption 1.
The total disturbancef is differentiable, and its derivative ḟ is described as ḟ=hx1,x2,Δ. The disturbance Δ 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 et evolves in the predefined bounds with a decreasing smooth function as follows:
(5)−ρt<et<ρt,∀t>0.The prescribed performance function (PPF)ρt is selected as [38]
(6)ρt=ρ0−ρ∞exp−κt+ρ∞,where ρ0>ρ∞>0 and κ>0. The maximum overshoot, steady-state error, and convergence rate will be limited by selecting the appropriate parameters ρ0, ρ∞, and κ, respectively. For a more intuitive insight of the concept, Figure 1 illustrates the schematic diagram of the aforementioned prescribed performance theory.Figure 1
The prescribed performance illustration.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 T⋅, i.e.,
(7)et=ρtTεt,where εt is the transformed tracking error, and T⋅ is smooth and strictly increasing which satisfies the conditions −1<Tε<1 and limε→+∞Tε=1,limε→−∞Tε=−1.Here, we design the error transformation functionT⋅ as
(8)Tε=expε−exp−εexpε+exp−ε.According to the property of the functionT⋅, we can inversely derive the error εt as follows:
(9)εt=T−1λ=12ln1+λ1−ln1−λ1,where λ=et/ρt is the normalized tracking error.Remark 2.
In the subsequent parts, the transformed errorεt will be used into the controller in place of the tracking error et to deal with the problem of control system design with prescribed performance constraint.


### 2.3. Control Objective

The control objective is the proposed novel controlleru for the ASV attitude system (3) that the state x1 track the desired command x1c accurately, and the tracking error e1=x1c−x1 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 statex1 can accurately track the desired command x1c with the unknown multiple disturbances(2)
The output tracking errore1 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


## 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:
(1)ϑ̇=ωz,ω̇z=MzJz,α=ϑ−θ,where the pitch moment Mz is defined as
(2)Mz=q¯SLm0+mzαα+mzα2α2+q¯SLmzδzδz,where elevator deflection δz is the control input, and m0, mzα, mzα2, andmzδz are the aerodynamic coefficients.For simplicity, defining the statex1,x2=ϑ,ωz and combining with (1) and (2), the ASV attitude control system can be described with the strict feedback state-space form:
(3)ẋ1=x2,ẋ2=f+bu,where
(4)f=q¯SLm0+mzαα+mzα2α2Jz+Δb=q¯SLmzδzJzu=δzHere, the termf is described as “total disturbance,” which consists of the unknown dynamic uncertainty and external disturbance.Remark 1.
The uncertainty termΔ is mainly the result of parameter uncertainties and external disturbances, e.g., the aerodynamic parameters perturbations and wind interference.Assumption 1.
The total disturbancef is differentiable, and its derivative ḟ is described as ḟ=hx1,x2,Δ. The disturbance Δ 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 et evolves in the predefined bounds with a decreasing smooth function as follows:
(5)−ρt<et<ρt,∀t>0.The prescribed performance function (PPF)ρt is selected as [38]
(6)ρt=ρ0−ρ∞exp−κt+ρ∞,where ρ0>ρ∞>0 and κ>0. The maximum overshoot, steady-state error, and convergence rate will be limited by selecting the appropriate parameters ρ0, ρ∞, and κ, respectively. For a more intuitive insight of the concept, Figure 1 illustrates the schematic diagram of the aforementioned prescribed performance theory.Figure 1
The prescribed performance illustration.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 T⋅, i.e.,
(7)et=ρtTεt,where εt is the transformed tracking error, and T⋅ is smooth and strictly increasing which satisfies the conditions −1<Tε<1 and limε→+∞Tε=1,limε→−∞Tε=−1.Here, we design the error transformation functionT⋅ as
(8)Tε=expε−exp−εexpε+exp−ε.According to the property of the functionT⋅, we can inversely derive the error εt as follows:
(9)εt=T−1λ=12ln1+λ1−ln1−λ1,where λ=et/ρt is the normalized tracking error.Remark 2.
In the subsequent parts, the transformed errorεt will be used into the controller in place of the tracking error et to deal with the problem of control system design with prescribed performance constraint.


## 2.3. Control Objective

The control objective is the proposed novel controlleru for the ASV attitude system (3) that the state x1 track the desired command x1c accurately, and the tracking error e1=x1c−x1 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 statex1 can accurately track the desired command x1c with the unknown multiple disturbances(2)
The output tracking errore1 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 Figure2, which mainly consists of three parts: prescribed performance control (PPC), linear extended state observer (LESO), and linear state error feedback (LSEF).Figure 2
The block diagram of the presented controller.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:
(10)ż1=z2+β1x1−z1,ż2=z3+β2x1−z1+bu,ż2=β3x1−z1,where z1, z2, and z3 are the observer states, and β1, β2, and β3 are the designed gains which can be chosen with the pole-placement method [24] as follows:
(11)β1=3ω0,β2=3ω02,β3=ω03,where ω0 is the observer bandwidth.With the properly selected gains,z1, z2 will converge to the system states x1 and x2, respectively, and z3 will accurately track the total disturbance term f, i.e., z1⟶x1,z2⟶x2,z3⟶f.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 errore~1 as
(12)e~1=x1c−z1,where x1c is the desired command and z1 is the estimation value of x1 obtained by LESO.Subsequently, the normalized errorλ1 is given by
(13)λ1=e~1ρ1t,where ρ1t is a similar PPF defined in (6), and it is expressed as
(14)ρ1t=ρ10−ρ1∞exp−κ1t+ρ1∞,whereρ10,ρ1∞, and κ1 are all positive constants.Then, the transformed error is obtained by(15)ε1=T−1λ1=12ln1+λ1−ln1−λ1.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:(16)u=u0+uf,where term u0 denotes a linear state error feedback control term that guarantees the system is asymptotically stable, and term uf 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 subitemu0 adopted the linear proportional and derivative (PD) control framework by introducing the transformed error ε1 of PPC which is designed as follows:
(17)u0=kpε1−kdz2b,where kp, kd are the controller gains and z2 is the estimation value of x2 obtained by LESO, and expression of b is given in (3).Meanwhile, the control subitemuf compensates the disturbances with the estimated value z3, which is given by
(18)uf=−z3b.Thus, the controlleru is obtained in terms of (16), (17), and (18) as follows:
(19)u=kpε1−kdz2−z3b.Note that the controller u0 introduces the transformed error ε1 instead of e~1 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 systemη̇t=Nηt+gt, where N is a n×n matrix, and ηt=η1t,η2t,⋯,ηnt, gt=g1t,g2t,⋯,gnt, if limt→∞gt=0 and N is Hurwitz, then limt→∞ηt=0 holds.This lemma has been proofed in detail which can be seen in [48].Lemma 2.
If the transformed errorεt can be controlled to be bounded, i.e., the condition εt≤εM holds where εM>0 is constant; then, the tracking error et can be controlled within a prescribed boundary, i.e., −ρt<et<ρt,∀t>0 holds.This lemma has been proofed in detail which can be seen in [38].Theorem 1.
For system (3) and LESO (10), if ḟ=hx1,x2,Δ satisfies the Lipschitz condition in the definition domain, there exist a constant ω0>0 such that the estimated states z1, z2, and z3 can converge to the state x1, x2, and f, respectively, i.e., the observer errors satisfy that limt→∞ezit=0,i=1,2,3, where the observer errors are defined as ez1=x1−z1, ez2=x2−z2, and ez3=f−z3.Proof.
Define the observer errors as(20)ez1=x1−z1,ez2=x2−z2,ez3=f−z3.From (3) and (10), the observer error dynamic can be given by
(21)ėz1=−3ω0ez1+ez2,ėz2=−3ω02ez1+ez3,ėz3=−ω03ez1+ḟ.Lete~=e~1,e~2,e~3=ez1,ez2/ω0,ez3/ω02 and ḟ=hx1,x2,Δ, then (21) can be rewritten as follows:
(22)e~̇=ω0A~e~+B~hx1,x2,Δω03,where A~=−310−300−100, B~=001.AsA~satisfies the Hurwitz stability, the condition PA~+A~TP=−I holds, where the matrix I is 3-order identity matrix and P is a positive definite Hermitian matrix and is expressed as
(23)P=1−12−1−121−12−1−124.Design the following Lyapunov function candidate(24)V1=e~TPe~.Substituting (23) into (24) yields
(25)V1=e~12+e~22+4e~32−e~1e~2−2e~1e~3−e~2e~3.The time derivative ofV1 along (25) is
(26)V̇1=2e~1e~̇1+2e~2e~̇2+8e~3e~̇3−e~̇1e~2−e~1e~̇2−2e~̇1e~3−2e~1e~̇3−e~̇2e~3−e~2e~̇3,=−ω0e~12+e~22+e~32+−2e~1−e~2+8e~3hx1,x2,Δω02≤−ω0e~2+−2e~1−e~2+8e~3hx1,x2,Δω02.The termhx1,x2,Δ satisfies [49].
(27)hx1,x2,Δ≤ke~,where k>0 is constant.Meanwhile, as the equation−2e~1−e~2+8e~3=2e~TPB~ holds, we can obtain that
(28)−2e~1−e~2+8e~3hx1,x2,Δω02≤2e~TPB~ke~ω02.In addition, the following inequality holds(29)PB~k2−2PB~k+1≥0.Thus, we obtain
(30)2e~TPB~ke~ω02≤Ke~2,where K=PBZk2+1/ω02.Based on (26), (28), and (30), we can obtain that
(31)V̇1≤−ω0−Ke~Z2.That is, ifω0>K such that V̇1<0. Therefore, limt→∞ezit=0,i=1,2,3 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 ḟ=hx1,x2,Δ satisfies the Lipschitz condition in definition domain, there exists constants ω0>0 and ωc>0 that can guarantee the system (3) to be asymptotically stable. Furthermore, the tracking error e1 can be maintained in a predefined set, i.e., the inequality −ρ1t<e1t<ρ1t holds.Proof.
Define the tracking error of system (3) as
(32)e1=x1c−x1,e2=x2c−x2.As the control commandx1c is assumed as a constant, we have x2c=ẋ1c=0. The time derivative of e1 and e2 along (3) is
(33)ė1=ẋ1c−ẋ1=−x2=e2,(34)ė2=ẋ2c−ẋ2=−ẋ2=−f−bu.Based on the expression of the transformed errorε1, it yields
(35)ε1=12ln1+λ1−ln1−λ1.Then, the functionln1+λ1 and ln1+λ1 can be derived using Taylor’s expansion as follows:
(36)ln1+λ1=λ1−12λ12+R2λ1,(37)ln1−λ1=−λ1+12λ12+R2λ1,where R2λ1 is the 2-order Taylor remainder. As the error λ1 is relatively small near zero, thus the R2λ1 can be neglected.Substituting (36) into (35), the transformed error ε1 can be approximately expressed as
(38)ε1=λ1=x1c−z1ρ1t.Substituting (38) into (19), the proposed controller (18) can be rewritten as
(39)u=kpx1c−z1/ρ1t−kdz2−z3b.Substituting (39) into (34) yields
(40)ė2=−f−bu=−kpx1c−z1ρt+kdz2+z3−f.Combining the expressions of the observer error and system error, we can obtain that(41)x1c−z1=e1+ez1,(42)z2=−e2+ez2,(43)z3−f=−ez3.Substituting (41) into (40) yields
(44)ė2=−kpρte1+ez1−kde2+ez2−ez3.From (33) and (44), the tracking error model is given by
(45)ė=Aee+Beez,where e=e1,e2T and ez=ez1,ez2T are the system tracking error vector and the observer error vector, respectively. The matrix Ae and Be are described as
(46)Ae=01−kpρ1t−kd,Be=000−kpρ1t−kd−1.Meanwhile, the prescribed performance functionρ1t satisfies
(47)limt→∞ρ1t=limt→∞ρ10−ρ1∞exp−κ1t+ρ1∞=ρ1∞.Thus, under the conditiont⟶∞, the matrix Ae and Be can be given as
(48)Ae=01−kpρ1∞−kd,Be=000−kpρ1∞−kd−1.According to Theorem1, we can obtain that limt→∞ez=0. Furthermore, if the characteristic polynomial s2+kds+kp/ρ1∞ satisfies the Routh criterion, here, we select s2+kds+kp/ρ1∞=s+ωc2, where ωc>0 is the controller bandwidth, i.e., kp=ρ∞ωc2,kd=2ωc such that the matrix Ae is Hurwitz. Based on Lemma 1, we can obtain that the limt→∞eit=0,i=1,2 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
(49)exp2ε1=1+λ11−λ1.Based on Lemma2, if the transformed errors ε1=T−1λ1 can be controlled to be bounded, i.e., ε1≤εM1 holds for positive constants εM1>0. This further implies
(50)−1<exp−εM1−1exp−εM1+1≤λ1≤expεM1−1expεM1+1<1.From the factλ1=e1t/ρ1t and ρ1t>0, the error e1 can be maintained within a predefined set, −ρ1t<e1t<ρ1t holds. This completes the proof.


## 3.1. Attitude Control System Design

The block diagram of the presented control system is depicted in Figure2, which mainly consists of three parts: prescribed performance control (PPC), linear extended state observer (LESO), and linear state error feedback (LSEF).Figure 2
The block diagram of the presented controller.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:
(10)ż1=z2+β1x1−z1,ż2=z3+β2x1−z1+bu,ż2=β3x1−z1,where z1, z2, and z3 are the observer states, and β1, β2, and β3 are the designed gains which can be chosen with the pole-placement method [24] as follows:
(11)β1=3ω0,β2=3ω02,β3=ω03,where ω0 is the observer bandwidth.With the properly selected gains,z1, z2 will converge to the system states x1 and x2, respectively, and z3 will accurately track the total disturbance term f, i.e., z1⟶x1,z2⟶x2,z3⟶f.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 errore~1 as
(12)e~1=x1c−z1,where x1c is the desired command and z1 is the estimation value of x1 obtained by LESO.Subsequently, the normalized errorλ1 is given by
(13)λ1=e~1ρ1t,where ρ1t is a similar PPF defined in (6), and it is expressed as
(14)ρ1t=ρ10−ρ1∞exp−κ1t+ρ1∞,whereρ10,ρ1∞, and κ1 are all positive constants.Then, the transformed error is obtained by(15)ε1=T−1λ1=12ln1+λ1−ln1−λ1.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:(16)u=u0+uf,where term u0 denotes a linear state error feedback control term that guarantees the system is asymptotically stable, and term uf 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 subitemu0 adopted the linear proportional and derivative (PD) control framework by introducing the transformed error ε1 of PPC which is designed as follows:
(17)u0=kpε1−kdz2b,where kp, kd are the controller gains and z2 is the estimation value of x2 obtained by LESO, and expression of b is given in (3).Meanwhile, the control subitemuf compensates the disturbances with the estimated value z3, which is given by
(18)uf=−z3b.Thus, the controlleru is obtained in terms of (16), (17), and (18) as follows:
(19)u=kpε1−kdz2−z3b.Note that the controller u0 introduces the transformed error ε1 instead of e~1 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 systemη̇t=Nηt+gt, where N is a n×n matrix, and ηt=η1t,η2t,⋯,ηnt, gt=g1t,g2t,⋯,gnt, if limt→∞gt=0 and N is Hurwitz, then limt→∞ηt=0 holds.This lemma has been proofed in detail which can be seen in [48].Lemma 2.
If the transformed errorεt can be controlled to be bounded, i.e., the condition εt≤εM holds where εM>0 is constant; then, the tracking error et can be controlled within a prescribed boundary, i.e., −ρt<et<ρt,∀t>0 holds.This lemma has been proofed in detail which can be seen in [38].Theorem 1.
For system (3) and LESO (10), if ḟ=hx1,x2,Δ satisfies the Lipschitz condition in the definition domain, there exist a constant ω0>0 such that the estimated states z1, z2, and z3 can converge to the state x1, x2, and f, respectively, i.e., the observer errors satisfy that limt→∞ezit=0,i=1,2,3, where the observer errors are defined as ez1=x1−z1, ez2=x2−z2, and ez3=f−z3.Proof.
Define the observer errors as(20)ez1=x1−z1,ez2=x2−z2,ez3=f−z3.From (3) and (10), the observer error dynamic can be given by
(21)ėz1=−3ω0ez1+ez2,ėz2=−3ω02ez1+ez3,ėz3=−ω03ez1+ḟ.Lete~=e~1,e~2,e~3=ez1,ez2/ω0,ez3/ω02 and ḟ=hx1,x2,Δ, then (21) can be rewritten as follows:
(22)e~̇=ω0A~e~+B~hx1,x2,Δω03,where A~=−310−300−100, B~=001.AsA~satisfies the Hurwitz stability, the condition PA~+A~TP=−I holds, where the matrix I is 3-order identity matrix and P is a positive definite Hermitian matrix and is expressed as
(23)P=1−12−1−121−12−1−124.Design the following Lyapunov function candidate(24)V1=e~TPe~.Substituting (23) into (24) yields
(25)V1=e~12+e~22+4e~32−e~1e~2−2e~1e~3−e~2e~3.The time derivative ofV1 along (25) is
(26)V̇1=2e~1e~̇1+2e~2e~̇2+8e~3e~̇3−e~̇1e~2−e~1e~̇2−2e~̇1e~3−2e~1e~̇3−e~̇2e~3−e~2e~̇3,=−ω0e~12+e~22+e~32+−2e~1−e~2+8e~3hx1,x2,Δω02≤−ω0e~2+−2e~1−e~2+8e~3hx1,x2,Δω02.The termhx1,x2,Δ satisfies [49].
(27)hx1,x2,Δ≤ke~,where k>0 is constant.Meanwhile, as the equation−2e~1−e~2+8e~3=2e~TPB~ holds, we can obtain that
(28)−2e~1−e~2+8e~3hx1,x2,Δω02≤2e~TPB~ke~ω02.In addition, the following inequality holds(29)PB~k2−2PB~k+1≥0.Thus, we obtain
(30)2e~TPB~ke~ω02≤Ke~2,where K=PBZk2+1/ω02.Based on (26), (28), and (30), we can obtain that
(31)V̇1≤−ω0−Ke~Z2.That is, ifω0>K such that V̇1<0. Therefore, limt→∞ezit=0,i=1,2,3 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 ḟ=hx1,x2,Δ satisfies the Lipschitz condition in definition domain, there exists constants ω0>0 and ωc>0 that can guarantee the system (3) to be asymptotically stable. Furthermore, the tracking error e1 can be maintained in a predefined set, i.e., the inequality −ρ1t<e1t<ρ1t holds.Proof.
Define the tracking error of system (3) as
(32)e1=x1c−x1,e2=x2c−x2.As the control commandx1c is assumed as a constant, we have x2c=ẋ1c=0. The time derivative of e1 and e2 along (3) is
(33)ė1=ẋ1c−ẋ1=−x2=e2,(34)ė2=ẋ2c−ẋ2=−ẋ2=−f−bu.Based on the expression of the transformed errorε1, it yields
(35)ε1=12ln1+λ1−ln1−λ1.Then, the functionln1+λ1 and ln1+λ1 can be derived using Taylor’s expansion as follows:
(36)ln1+λ1=λ1−12λ12+R2λ1,(37)ln1−λ1=−λ1+12λ12+R2λ1,where R2λ1 is the 2-order Taylor remainder. As the error λ1 is relatively small near zero, thus the R2λ1 can be neglected.Substituting (36) into (35), the transformed error ε1 can be approximately expressed as
(38)ε1=λ1=x1c−z1ρ1t.Substituting (38) into (19), the proposed controller (18) can be rewritten as
(39)u=kpx1c−z1/ρ1t−kdz2−z3b.Substituting (39) into (34) yields
(40)ė2=−f−bu=−kpx1c−z1ρt+kdz2+z3−f.Combining the expressions of the observer error and system error, we can obtain that(41)x1c−z1=e1+ez1,(42)z2=−e2+ez2,(43)z3−f=−ez3.Substituting (41) into (40) yields
(44)ė2=−kpρte1+ez1−kde2+ez2−ez3.From (33) and (44), the tracking error model is given by
(45)ė=Aee+Beez,where e=e1,e2T and ez=ez1,ez2T are the system tracking error vector and the observer error vector, respectively. The matrix Ae and Be are described as
(46)Ae=01−kpρ1t−kd,Be=000−kpρ1t−kd−1.Meanwhile, the prescribed performance functionρ1t satisfies
(47)limt→∞ρ1t=limt→∞ρ10−ρ1∞exp−κ1t+ρ1∞=ρ1∞.Thus, under the conditiont⟶∞, the matrix Ae and Be can be given as
(48)Ae=01−kpρ1∞−kd,Be=000−kpρ1∞−kd−1.According to Theorem1, we can obtain that limt→∞ez=0. Furthermore, if the characteristic polynomial s2+kds+kp/ρ1∞ satisfies the Routh criterion, here, we select s2+kds+kp/ρ1∞=s+ωc2, where ωc>0 is the controller bandwidth, i.e., kp=ρ∞ωc2,kd=2ωc such that the matrix Ae is Hurwitz. Based on Lemma 1, we can obtain that the limt→∞eit=0,i=1,2 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
(49)exp2ε1=1+λ11−λ1.Based on Lemma2, if the transformed errors ε1=T−1λ1 can be controlled to be bounded, i.e., ε1≤εM1 holds for positive constants εM1>0. This further implies
(50)−1<exp−εM1−1exp−εM1+1≤λ1≤expεM1−1expεM1+1<1.From the factλ1=e1t/ρ1t and ρ1t>0, the error e1 can be maintained within a predefined set, −ρ1t<e1t<ρ1t 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:
(51)uLADC=kpx1c−z1−kdz2−z3b,where z1, z2, and z3 is obtained by LESO.


### 4.2. Flight Conditions

The vehicle is assumed to maintain altitudeH=12 km cruise at constant velocity V=1000 m/s such that the flight path angle is θ=0deg, i.e., the variation of the angle of attack α is the same as that of the pitch angle ϑ based on the system (1). The aerodynamic coefficients are m0=0.0005, mzα=−0.018deg−1, mzα2=−0.003deg−2, and mzδz=−0.035deg−1. The initial conditions for the ASV are set as ϑ0=0deg, ωz0=0deg/s.


### 4.3. Controller Parameters

The relevant parameters of PPF is designed asρ10=1.5, ρ1∞=0.05, and κ1=5, i.e., the PPF is selected as ρ1t=1.5−0.05exp−5t+0.05. The observer bandwidth ω0 is set as ω0=40, and controller bandwidth ωc is designed as ωc=10. Then, the observer gains β1, β2, and β3 and controller gains kp, kd can be obtained according to the correspondence with the bandwidth ω0 and ωc.


### 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)tc of the tracking error (TE)The convergence timetc is assumed as the corresponding time that the tracking error e1 is e1≤10−6 and continues at least 10 sampling periods.(2)
Average value (AV)μe of the TE
(52)μe=1n∑i=1ne1i,where n is the sample point number(3)
Standard deviation (SD)σe of the TE
(53)σe=1n∑i=1ne1i−μe2(4)
Amount of control consumption (ACC)Q(54)Q=∫t0tfu2dt,where t0 and tf are the start time and end time, respectively, u 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 termΔ in system (3) is Δ=0.The comparative results are summarized in Table1 and Figures 3–10.Table 1
The comparative simulation results of Case1.
IndexLADRCLADRC-PPCRatioCTtc2.65 s0.99 s2.57 : 1AVμe0.07 deg0.03 deg2.33 : 1SDσe0.18 deg0.14 deg1.28 : 1ACCQ1.261.301 : 1.03Figure 3
Pitch angle.Figure 4
Tracking error of pitch angle.Figure 5
Pitch angular rate.Figure 6
Control input signal-elevator deflection.Figure 7
Estimation off with LADRC.Figure 8
Estimation off with LADRC-PPC.Figure 9
Estimation error off with LADRC.Figure 10
Estimation error off with LADRC-PPC.As depicted in Figures3 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 termΔ is mainly considered two parts: (1) the model uncertainties Δ1, and the aerodynamic coefficients are presumed to decrease by 10% on the basis of standard values, the density of air ρ is perturbed to be +5% and (2) external disturbance Δ2, and it is considered an abrupt one Δ2=0.1deg/s2t≥3, i.e., the uncertainty term Δ is set as follows
(55)Δ=Δ1t<3Δ1+0.1t≥3deg/s2.The comparative results are summarized in Table2 and Figures 11–18.Table 2
The comparative simulation results of Case2.
IndexLADRCLADRC-PPCRatioCTtc2.74 s1.10s2.49 : 1AVμe0.07 deg0.03 deg2.33 : 1SDσe0.18 deg0.14 deg1.28 : 1ACCQ1.061.101 : 1.04Figure 11
Pitch angle.Figure 12
Tracking error of pitch angle.Figure 13
Pitch angular rate.Figure 14
Control input signal-elevator deflection.Figure 15
Estimation off with LADRC.Figure 16
Estimation off with LADRC-PPC.Figure 17
Estimation error off with LADRC.Figure 18
Estimation error off with LADRC-PPC.From Figure11, 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 ofN=1000 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 parameterJz and ρ is -10% ~ +10% of the standard value, which follows the random variation with normal distributionFigures19–24 depicts the 1000-run Monte Carlo simulation results, and Table 3 illustrates the corresponding statistical results including the expectation value μ and average variance σ of the above indexes CT tc, AV μe, and ACC Q.Figure 19
CT of Monte Carlo.Figure 20
CT of Monte Carlo.Figure 21
AV of Monte Carlo.Figure 22
AV of Monte Carlo.Figure 23
ACC of Monte Carlo.Figure 24
ACC of Monte Carlo.Table 3
The statistical results of the Monte Carlo simulations.
Statistical dataLADRCLADRC-PPCtcμ2.72571.0710σ0.10220.0750μeμ0.06410.0374σ0.00150.0007Qμ1.22701.2880σ0.22530.2353The Monte Carlo simulation results are depicted in Figures19–24, and Table 3 lists the statistical results of 1000 Monte Carlo simulations which included the expectation μ and average variance σ of the concerned indexes CT tc, AV μe, and ACC Q.Figures19 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 tcwith 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 μe exhibit the similar regularity as shown in Figures 21 and 22. The distributions of the amount of control consumption Q with the above controllers are illustrated in Figures 23 and 24, and the expectation of Q 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.


## 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:
(51)uLADC=kpx1c−z1−kdz2−z3b,where z1, z2, and z3 is obtained by LESO.


## 4.2. Flight Conditions

The vehicle is assumed to maintain altitudeH=12 km cruise at constant velocity V=1000 m/s such that the flight path angle is θ=0deg, i.e., the variation of the angle of attack α is the same as that of the pitch angle ϑ based on the system (1). The aerodynamic coefficients are m0=0.0005, mzα=−0.018deg−1, mzα2=−0.003deg−2, and mzδz=−0.035deg−1. The initial conditions for the ASV are set as ϑ0=0deg, ωz0=0deg/s.


## 4.3. Controller Parameters

The relevant parameters of PPF is designed asρ10=1.5, ρ1∞=0.05, and κ1=5, i.e., the PPF is selected as ρ1t=1.5−0.05exp−5t+0.05. The observer bandwidth ω0 is set as ω0=40, and controller bandwidth ωc is designed as ωc=10. Then, the observer gains β1, β2, and β3 and controller gains kp, kd can be obtained according to the correspondence with the bandwidth ω0 and ωc.


## 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)tc of the tracking error (TE)The convergence timetc is assumed as the corresponding time that the tracking error e1 is e1≤10−6 and continues at least 10 sampling periods.(2)
Average value (AV)μe of the TE
(52)μe=1n∑i=1ne1i,where n is the sample point number(3)
Standard deviation (SD)σe of the TE
(53)σe=1n∑i=1ne1i−μe2(4)
Amount of control consumption (ACC)Q(54)Q=∫t0tfu2dt,where t0 and tf are the start time and end time, respectively, u 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 termΔ in system (3) is Δ=0.The comparative results are summarized in Table1 and Figures 3–10.Table 1
The comparative simulation results of Case1.
IndexLADRCLADRC-PPCRatioCTtc2.65 s0.99 s2.57 : 1AVμe0.07 deg0.03 deg2.33 : 1SDσe0.18 deg0.14 deg1.28 : 1ACCQ1.261.301 : 1.03Figure 3
Pitch angle.Figure 4
Tracking error of pitch angle.Figure 5
Pitch angular rate.Figure 6
Control input signal-elevator deflection.Figure 7
Estimation off with LADRC.Figure 8
Estimation off with LADRC-PPC.Figure 9
Estimation error off with LADRC.Figure 10
Estimation error off with LADRC-PPC.As depicted in Figures3 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 termΔ is mainly considered two parts: (1) the model uncertainties Δ1, and the aerodynamic coefficients are presumed to decrease by 10% on the basis of standard values, the density of air ρ is perturbed to be +5% and (2) external disturbance Δ2, and it is considered an abrupt one Δ2=0.1deg/s2t≥3, i.e., the uncertainty term Δ is set as follows
(55)Δ=Δ1t<3Δ1+0.1t≥3deg/s2.The comparative results are summarized in Table2 and Figures 11–18.Table 2
The comparative simulation results of Case2.
IndexLADRCLADRC-PPCRatioCTtc2.74 s1.10s2.49 : 1AVμe0.07 deg0.03 deg2.33 : 1SDσe0.18 deg0.14 deg1.28 : 1ACCQ1.061.101 : 1.04Figure 11
Pitch angle.Figure 12
Tracking error of pitch angle.Figure 13
Pitch angular rate.Figure 14
Control input signal-elevator deflection.Figure 15
Estimation off with LADRC.Figure 16
Estimation off with LADRC-PPC.Figure 17
Estimation error off with LADRC.Figure 18
Estimation error off with LADRC-PPC.From Figure11, 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 ofN=1000 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 parameterJz and ρ is -10% ~ +10% of the standard value, which follows the random variation with normal distributionFigures19–24 depicts the 1000-run Monte Carlo simulation results, and Table 3 illustrates the corresponding statistical results including the expectation value μ and average variance σ of the above indexes CT tc, AV μe, and ACC Q.Figure 19
CT of Monte Carlo.Figure 20
CT of Monte Carlo.Figure 21
AV of Monte Carlo.Figure 22
AV of Monte Carlo.Figure 23
ACC of Monte Carlo.Figure 24
ACC of Monte Carlo.Table 3
The statistical results of the Monte Carlo simulations.
Statistical dataLADRCLADRC-PPCtcμ2.72571.0710σ0.10220.0750μeμ0.06410.0374σ0.00150.0007Qμ1.22701.2880σ0.22530.2353The Monte Carlo simulation results are depicted in Figures19–24, and Table 3 lists the statistical results of 1000 Monte Carlo simulations which included the expectation μ and average variance σ of the concerned indexes CT tc, AV μe, and ACC Q.Figures19 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 tcwith 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 μe exhibit the similar regularity as shown in Figures 21 and 22. The distributions of the amount of control consumption Q with the above controllers are illustrated in Figures 23 and 24, and the expectation of Q 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.



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