# Prescribed Performance Tracking Control for Hypersonic Flight Vehicles with Model Uncertainties

Authors: Siyuan Zhao; Xiaobing Li; Xiangwei Bu; Dongyang Zhang
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

This paper proposes a novel prescribed performance tracking control for a hypersonic flight vehicle (HFV) with model uncertainties. Firstly, a HFV longitudinal motion model is decomposed into a velocity subsystem and an altitude subsystem. Meanwhile, considering the uncertainties of the model, the velocity subsystem and altitude subsystem are directly expressed as the forms with unknown nonaffine functions. Secondly, a novel performance function without initial error is proposed for limiting the tracking error into a prescribed range. Then, for the altitude subsystem, the control objective is changed by model transformation and the prescribed performance backstepping controller is designed. For the velocity subsystem, a prescribed performance proportional-integral controller is proposed which has better engineering practicability. The designed controller is not only simple in form but also has few calculating parameters. Finally, the simulation results show that the proposed controller has good practicability.


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## Body

## 1. Introduction

Near space usually refers to the 20~100 km airspace from the horizontal plane [1]. This airspace is located between the traditional concept of “aviation” and “aerospace,” across the stratosphere and ionosphere, which is rarely involved in the past aircraft airspace. Thus, the development prospects are very broad [2, 3]. A flight vehicle flying in near space and with hypersonic speed is called a hypersonic flight vehicle (HFV). HFV is considered as a key step towards providing a promising and affordable technology to meet commercial and military goals [4]. Compared with traditional aircraft, HFV has stronger nonlinear and coupling characteristics, more serious elastic vibration, and more strict control constraints. The complex dynamic characteristics of HFV will undoubtedly bring great challenges to the design of its controller, which makes the HFV control problem a hot research topic [5–7].The modeling and control design of HFV are mainly carried out on its longitudinal motion model. On one hand, its longitudinal motion model for flight control is complicated enough; on the other hand, considering the scramjet’s sensitivity to flight attitude and fuel saving, lateral maneuver should be avoided as much as possible during actual cruise flight [8]. Due to the difficulty to establish a precise motion model for HFV, Ref. [9] investigates a novel model-free controller for the HFV longitudinal motion model. Since the HFV model has fast time variability, strong coupling, and highly nonlinear and uncertain parameters, a sliding-mode decoupling attitude controller based on parametric commands is proposed which takes these features into account [10]. In order to facilitate the design of the controller, the input and output linearized high-order model of HFV is transformed into a multivariable second-order system model by introducing auxiliary error variables. Meanwhile, in order to ensure the actual finite-time stability of the sliding-mode manifold, an adaptive fast nonsingular terminal sliding-mode controller is designed for HFV with an unknown upper limit of disturbance [11].The basic idea of a backstepping control design is to decompose the complex nonlinear system into subsystems with no more than orders of the system, and then design a Lyapunov function and an intermediate virtual control quantity for each subsystem, then push it to the whole system, and complete the design of the whole control law by integrating them [12]. Backstepping control is suitable for uncertain nonlinear systems with state linearization or strict parameter feedback, especially in the field of aeronautics and astronautics [13]. At the same time, intelligent control has good performance in dealing with model uncertain control problems [14–16]. Therefore, many studies have combined the idea of an intelligent control and backstepping control method to solve the HFV control problem. Ref. [17] discusses the design of HFV adaptive fuzzy backstepping tracking control with actuator constraints and applies the fuzzy logic system to approximate analysis of the total uncertainty of the fuzzy fractal method model. Based on the control-oriented model of HFV, an adaptive backstepping controller and a dynamic inverse controller are designed for the altitude subsystem and velocity subsystem, respectively, in Ref. [18]. Meanwhile, in order to avoid the complexity problems caused by the repeated derivation of virtual control variables by traditional synchronization control, the dynamic surface control technique and synchronization control method are involved, and a novel second-order sliding-mode-based integral filter is used to replace the traditional first-order filter. In order to compensate for the dead zone effect of HFV and reduce the computation load, an improved dead zone smooth inverse is proposed, on which basis an input nonlinear precompensator is designed to deal with input saturation and dead zone nonlinearity [19].The existing control methods pay less attention to the overall control performance of HFV, but too much attention is paid to the steady progress of the control system [20, 21]. The prescribed performance control method requires that the system can meet the expected dynamic performance and steady performance at the same time, so as to improve the integrity of the system for the control goal, which has extremely important theoretical and practical significance [22]. Different from the traditional nonaffine model that requires nonaffine functions to be differentiable, Ref. [23] uses the semidecomposed nonaffine model to design an improved performance controller with a simple structure and low computational complexity on the basis of the back-extrapolation technique.Consistent with a quadrotor unmanned aerial vehicle system proposed in Ref. [24], the actual control model of HFV is also uncertain. Few examples of previous studies have applied HFV’s uncertain nonaffine model in the research of prescribed performance tracking control. In order to improve the overall control performance of HFV with an unknown model, a prescribed performance tracking control is designed in this paper. Firstly, the HFV longitudinal motion model is decomposed into a velocity subsystem and an altitude subsystem. At the same time, considering the uncertainty of the model, the velocity subsystem and the altitude subsystem are directly expressed as the forms with unknown nonaffine functions. Secondly, a new performance function without initial error is proposed to limit the tracking error within the specified range. Then, the control target of the altitude subsystem is changed by model transformation, and the corresponding backstepping controller is designed. Unlike previous studies, the backstepping controller designed in this paper reduces the steps and computational load. A prescribed performance Proportional-Integral (PI) controller is proposed for the velocity subsystem. The controller is simple in form and has few calculating parameters. Simulation results show that the controller has good practicability. The main contributions of this paper are summarized as follows:
(1)
The control method designed in this paper is based on the uncertain nonaffine models of HFV, which are closer to the actual flight situation than previous studies
(2)
The novel prescribed performance function is introduced to ensure transient performance and steady-state accuracy without initial error
(3)
The uncertainties in the HFV flight control are realized by using the backstepping control method to meet the requirements of matching conditions. There are less steps in the backstepping design and low computational load. Meanwhile, a PI controller, which is more commonly used in engineering practice, is applied in the velocity subsystem
(4)
Finally, the simulation results are compared with several methods proposed in existing references under different conditionsThe rest of this paper is organized as follows. The HFV model description and transformation are described in Section2, and the control laws are addressed in Section 3. In Section 4, the simulation results are presented. Finally, Section 5 provides the conclusion of this paper.


## 2. Model Description and Transformation

### 2.1. Model Description

As is shown to us, the longitudinal models of the controller design for HFV are usually considered as follows. The winged-cone rigid-body model of NASA is mostly used in the early Ref. [25]. The disadvantage of this model is that the elastic effect of HFV during flight is not taken into account. In order to consider the influence of the elastic state, the integrated analytical model [26] and the improved analytical model [27, 28] are proposed. On the basis of these two models, Parker, a researcher in the United States Air Force Research Office, established the HFV control-oriented parameter fitting model through the Hooke law and the Lagrange theorem [7]. In this study, we consider the longitudinal model of HFV developed by Parker. The geometry and force map of the HFV models are shown in Figure 1.
(1)V̇=Tcosθ−γm−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=LmV+Tsinθ−γmV−gcosγ,(4)θ̇=Q,(5)Q̇=M+ψ~1η¨1+ψ~2η¨2Iyy,(6)k1η¨1=−2ζ1ω1η̇1−ω12η1+N1−ψ¨1MIyy−ψ~1ψ~2η¨2Iyy,(7)k2η¨2=−2ζ2ω2η̇2−ω22η2+N2−ψ~2MIyy−ψ~2ψ~1η¨1Iyy,where five rigid-body states V, h, γ, θ, and Q represent velocity, altitude, flight-path angle, pitch angle, and pitch rate, respectively. η1 and η2 are the elastic states. T, D, L, M, N1, and N2 mean the thrust force, the drag force, the lift force, the pitching moment, the first generalized force, and the second generalized force which are defined, respectively, as follows [7]:
(8)T≈β1h,q¯Φα3+β2h,q¯α3+β3h,q¯Φα2+β4h,q¯α2+β5h,q¯Φα+β6h,q¯α+β7h,q¯Φ+β8h,q¯,D≈q¯SCDα2α2+q¯SCDαα+q¯SCDδe2δe2+q¯SCDδeδe+q¯SCD0,M≈zTT+q¯Sc¯CM,αα2α2+q¯Sc¯CM,ααα+q¯Sc¯CM,α0+q¯Sc¯ceδe,L≈q¯SCLαα+q¯SCLδeδe+q¯SCL0,N1=N1α2α2+N1αα+N10,N2=N2α2α2+N2αα+N2δeδe+N20,q¯=ρ¯V22,ρ¯=ρ¯0exp−h−h0hs,where the control inputs Φ and δe stand for fuel equivalence ratio and elevator angular deflection, and they occur implicitly in (1), (2), (3), (4), (5), (6), and (7). Should the readers want to know more about parameters and aerodynamic parameters in the above equations, they could refer to Ref. [7].Figure 1
Geometry and force map of HFV model.Usually, only the rigid-body statesV, h, γ, θ and Q of HFV are measurable which can be utilized in the design of control laws. Also, the elastic states η1 and η2 are considered as external disturbances when the controller is designed because they are not measurable.Definition 1 [29, 30].
A continuous functionρt that satisfies the following two conditions simultaneously is called a performance function:
(1)
ρt is continuously differentiable, bounded, strictly positive, and has a decreasing function of time
(2)
limt→∞ρt=ρ∞>0.According toDefinition 1, this paper chooses the following function as the performance function:
(9)ρt=cschlt+ρ0+ρ∞,where l, ρ0, and ρ∞ are positive parameters. l is relative to the convergence rate of the tracking error. ρ0 is relative to the boundedness of the tracking error. ρ0=cschρ0+ρ∞ represents the maximum overshoot of the tracking error. The convergence interval of the tracking error et is defined as
(10)−ρt<et<ρt.Obviously, it is impossible to design the controller directly from inequality (10). In this paper, we introduce an error transformation function as follows:
(11)Tε=eεt−e−εteεt+e−εt,where εt is the transform error.


### 2.2. Model Transformation

According to the timescale principle in [31], velocity has slower dynamics compared with altitude angles. Thus, the HFV motion model can be decomposed into a velocity subsystem and an altitude subsystem.A velocity subsystem can be represented as follows:(12)V̇=f0V,Φ,where f0V,Φ is a continuous unknown differentiable nonaffine function.The altitude tracking errorh~ is defined as
(13)h~=h−href.The reference command ofγ is designed as
(14)γd=arcsin−khεht+ḣrefV,where kh>0 is the design parameter. Meanwhile, εht=lnh~/ρht+1/1−h~/ρht is the altitude transform error and ρht=cschlht+ρh0+ρh∞.Ifγ→γd, the corresponding dynamics for εht is derived as
(15)ε̇ht+khεht=0.Obviously, ifεht is bounded, the transient performance and steady-state performance of h~ are both guaranteed [32, 33]. Thus, the control objective of the altitude subsystem is changed into γ→γd.In this way, the subsequent design goal is to letγ→γd. Then, the altitude subsystem is changed into
(16)ẋ1=f1x1,x2+x2,ẋ2=f2x+x3,ẋ3=f3x,δe+δe,where f1x1,x2, f2x, and f3x,δe are continuous unknown differentiable nonaffine functions, x=x1,x2,x3T, where x1=γ, x2=θ, and x3=Q.According to the prescribed performance control method, we consider the following change of coordinate:(17)γ~=x1−γd,where γ~ is the error flight-path angle. The time derivative of (17) is as follows:
(18)γ~̇=f1x1,x2+x2−γ̇d.Based onet=ρtTεt, the time derivative of the transform error ε1t of the flight-path angle is as follows:
(19)ε̇1t=γ~−ρ̇1tTε1tρ1t∂Tε1t/∂ε1t.Meanwhile,ε1t=lnγ~/ρ1t+1/1−γ~/ρ1t and ρ1t=cschl1t+ρ10+ρ1∞. Let Γ=−ρ̇1tTε1t/ρ1t∂Tε1t/∂ε1t and Λ=1/ρ1t∂Tε1t/∂ε1t, then the above altitude subsystem is modified as
(20)ε̇1t=Γ+Λf1x1,x2+x2−γ̇d,ẋ2=f2x+x3,ẋ3=f3x,δe+δe.


## 2.1. Model Description

As is shown to us, the longitudinal models of the controller design for HFV are usually considered as follows. The winged-cone rigid-body model of NASA is mostly used in the early Ref. [25]. The disadvantage of this model is that the elastic effect of HFV during flight is not taken into account. In order to consider the influence of the elastic state, the integrated analytical model [26] and the improved analytical model [27, 28] are proposed. On the basis of these two models, Parker, a researcher in the United States Air Force Research Office, established the HFV control-oriented parameter fitting model through the Hooke law and the Lagrange theorem [7]. In this study, we consider the longitudinal model of HFV developed by Parker. The geometry and force map of the HFV models are shown in Figure 1.
(1)V̇=Tcosθ−γm−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=LmV+Tsinθ−γmV−gcosγ,(4)θ̇=Q,(5)Q̇=M+ψ~1η¨1+ψ~2η¨2Iyy,(6)k1η¨1=−2ζ1ω1η̇1−ω12η1+N1−ψ¨1MIyy−ψ~1ψ~2η¨2Iyy,(7)k2η¨2=−2ζ2ω2η̇2−ω22η2+N2−ψ~2MIyy−ψ~2ψ~1η¨1Iyy,where five rigid-body states V, h, γ, θ, and Q represent velocity, altitude, flight-path angle, pitch angle, and pitch rate, respectively. η1 and η2 are the elastic states. T, D, L, M, N1, and N2 mean the thrust force, the drag force, the lift force, the pitching moment, the first generalized force, and the second generalized force which are defined, respectively, as follows [7]:
(8)T≈β1h,q¯Φα3+β2h,q¯α3+β3h,q¯Φα2+β4h,q¯α2+β5h,q¯Φα+β6h,q¯α+β7h,q¯Φ+β8h,q¯,D≈q¯SCDα2α2+q¯SCDαα+q¯SCDδe2δe2+q¯SCDδeδe+q¯SCD0,M≈zTT+q¯Sc¯CM,αα2α2+q¯Sc¯CM,ααα+q¯Sc¯CM,α0+q¯Sc¯ceδe,L≈q¯SCLαα+q¯SCLδeδe+q¯SCL0,N1=N1α2α2+N1αα+N10,N2=N2α2α2+N2αα+N2δeδe+N20,q¯=ρ¯V22,ρ¯=ρ¯0exp−h−h0hs,where the control inputs Φ and δe stand for fuel equivalence ratio and elevator angular deflection, and they occur implicitly in (1), (2), (3), (4), (5), (6), and (7). Should the readers want to know more about parameters and aerodynamic parameters in the above equations, they could refer to Ref. [7].Figure 1
Geometry and force map of HFV model.Usually, only the rigid-body statesV, h, γ, θ and Q of HFV are measurable which can be utilized in the design of control laws. Also, the elastic states η1 and η2 are considered as external disturbances when the controller is designed because they are not measurable.Definition 1 [29, 30].
A continuous functionρt that satisfies the following two conditions simultaneously is called a performance function:
(1)
ρt is continuously differentiable, bounded, strictly positive, and has a decreasing function of time
(2)
limt→∞ρt=ρ∞>0.According toDefinition 1, this paper chooses the following function as the performance function:
(9)ρt=cschlt+ρ0+ρ∞,where l, ρ0, and ρ∞ are positive parameters. l is relative to the convergence rate of the tracking error. ρ0 is relative to the boundedness of the tracking error. ρ0=cschρ0+ρ∞ represents the maximum overshoot of the tracking error. The convergence interval of the tracking error et is defined as
(10)−ρt<et<ρt.Obviously, it is impossible to design the controller directly from inequality (10). In this paper, we introduce an error transformation function as follows:
(11)Tε=eεt−e−εteεt+e−εt,where εt is the transform error.


## 2.2. Model Transformation

According to the timescale principle in [31], velocity has slower dynamics compared with altitude angles. Thus, the HFV motion model can be decomposed into a velocity subsystem and an altitude subsystem.A velocity subsystem can be represented as follows:(12)V̇=f0V,Φ,where f0V,Φ is a continuous unknown differentiable nonaffine function.The altitude tracking errorh~ is defined as
(13)h~=h−href.The reference command ofγ is designed as
(14)γd=arcsin−khεht+ḣrefV,where kh>0 is the design parameter. Meanwhile, εht=lnh~/ρht+1/1−h~/ρht is the altitude transform error and ρht=cschlht+ρh0+ρh∞.Ifγ→γd, the corresponding dynamics for εht is derived as
(15)ε̇ht+khεht=0.Obviously, ifεht is bounded, the transient performance and steady-state performance of h~ are both guaranteed [32, 33]. Thus, the control objective of the altitude subsystem is changed into γ→γd.In this way, the subsequent design goal is to letγ→γd. Then, the altitude subsystem is changed into
(16)ẋ1=f1x1,x2+x2,ẋ2=f2x+x3,ẋ3=f3x,δe+δe,where f1x1,x2, f2x, and f3x,δe are continuous unknown differentiable nonaffine functions, x=x1,x2,x3T, where x1=γ, x2=θ, and x3=Q.According to the prescribed performance control method, we consider the following change of coordinate:(17)γ~=x1−γd,where γ~ is the error flight-path angle. The time derivative of (17) is as follows:
(18)γ~̇=f1x1,x2+x2−γ̇d.Based onet=ρtTεt, the time derivative of the transform error ε1t of the flight-path angle is as follows:
(19)ε̇1t=γ~−ρ̇1tTε1tρ1t∂Tε1t/∂ε1t.Meanwhile,ε1t=lnγ~/ρ1t+1/1−γ~/ρ1t and ρ1t=cschl1t+ρ10+ρ1∞. Let Γ=−ρ̇1tTε1t/ρ1t∂Tε1t/∂ε1t and Λ=1/ρ1t∂Tε1t/∂ε1t, then the above altitude subsystem is modified as
(20)ε̇1t=Γ+Λf1x1,x2+x2−γ̇d,ẋ2=f2x+x3,ẋ3=f3x,δe+δe.


## 3. Controller Design

The control objective is to devise prescribed performance tracking control lawsΦ and δe, such that V and h can accurately track their reference inputs Vref and href. Meanwhile, the control laws need to guarantee tracking errors with a prescribed transient. To achieve the control goal, the following lemmas and assumption need to be considered.Lemma 1 [34].
The unknown nonlinear functionf1, f2, and f3 satisfy the following inequality:
(21)fix¯i≤x1+⋯+xiϒix1,⋯,xi,i=1,⋯,3,where ϒix1,⋯,xi are the known C1 functions.Assumption 1 [35, 36].
For the unknown velocity subsystem (12) and altitude subsystem (16), the desired trajectory and its derivative are bounded.Lemma 2 [37, 38].
Supposing the nonnegative definite functiongt satisfies
(22)dgtdt=−g1gtg2,where g1>0 and 0<g2<1 are design constants. Then, by solving the above differential equation, we can get
(23)gt=g01−g2−g11−g2t1/1−g2,t∈0,t0,0,t∈t0,+∞,with t0=g01−g2/g11−g2.


### 3.1. Altitude Control Design

By considering equations (10), (11), and (17), there are x1≤ρ1tε1t+γd and xi≤ei+αi−1 with ei=xi−αi−1, where αi−1 is the virtual controller.Step 1.
It follows that fromLemma 1 and Young’s inequality, there is
(24)ε1tΛf1≤Λε1tx1ϒ1≤Λε1tϒ1ρ1tε1t+γd≤Λε1t2ϒ1ρ1t+Λϒ1ε1tγd2+14.
Choose the following Lyapunov function:(25)L1=12ε1t2.
The time derivative of (25) is as follows:
(26)L1≤ε1tΓ+Λf1+e2+α1−γ̇d≤ε1tΓ+Λe2+α1−γ̇d+ε1tΛf1≤ε1tΓ+Λe2+α1−γ̇d+ε1t2ρ1tΛϒ1+Λϒ1ε1tγd2+14=ε1tΓ+Λe2+α1−γ̇d+ϒ1ρ1tε1t+Λϒ12γd2ε1t+14.
Thus, choose the virtual control law as(27)α1=1Λ−k1ε1t−Γ−Λϒ1ρ1tε1t+Λϒ12γd2ε1t−γ̇d,where k1 is the design parameter. Substituting (27) into (26) results in
(28)L̇1≤−k1ε1t2+Λε1te2.Step 2.
Similar to Step1, there is
(29)e2f2=e2ϒ2x1+x2≤e2ϒ2ρ1tε1t+γd+e2+α1=ϒ2e22+e2ϒ2ρ1tε1t+γd+α1≤ϒ2e22+e2ϒ2ρ1tε1t+γd+α12+14.
Choose the following Lyapunov function:(30)L2=L1+12e22.
The time derivative of (30) is as follows:
(31)L̇2=L̇1+e2f2+x3−α̇1≤−k1ε1t2+ε1tΛe2+14+e2e3+x3−α̇1+ϒ2e22+e22ϒ2ρ1tε1t+γd+α12+14=−k1ε1t2+12+e2ε1tΛ+e3+α2−α̇1+ϒ2e2+e2ϒ2ρ1tε1t+γd+α12.
Thus, choose the virtual control law as(32)α2=−k2e2−ε1tΛ−ϒ2e2−e2ϒ2ρ1tε1t+γd+α12+α̇1,where k2 is the design parameter. Substituting (32) into (31) results in
(33)L̇2≤−k1ε1t2−k2e22+e2e3+12.Step 3.
Similar to Step2, there is
(34)e3f3≤ϒ3e32+e3ϒ3ρ1tε1t+γd+e2+α1+α22+14.
Choose the following Lyapunov function:(35)L3=L2+12e32.
The time derivative of (35) is as follows:
(36)L̇3=L̇2+e3f3+δe−α̇2≤−k1ε1t2−k2e22−k3e32+34+e3e3ϒ3ρ1tε1t+γd+e2+α1+α22+e2+ϒ3e3+δe−α̇2.
Thus, choose the control law as(37)δe=−k3e3−e2−ϒ3e3−e3ϒ3ρ1tε1t+γd+e2+α1+α22+α̇2,where k3 is the design parameter. It follows from (37) and (36) that (38) holds
(38)L̇3≤−k1ε1t2−k2e22+k3e32+34=−kL3+34,where k=min2k1,2k2,2k3. Then, solving the above differential equation, there is
(39)L3≤e−ktL30−34+34.Obviously, from Lemma2, transform error ε1t and tracking errors e2 and e3 are bounded. Furthermore, all the signals in the closed-loop are bounded. Thus, the transient performance and steady-state accuracy of the altitude subsystem are guaranteed.


### 3.2. Velocity Control Design

A PI controller is designed for the velocity subsystem because the velocity dynamics is much simpler than the altitude dynamic change.(40)Φ=−kV1ε2t−kV2∫0tε2tdt+Φ0,with ε2t=lnV~/ρ2t+1/1−V~/ρ2t, V~=V−Vref, and ρ2t=cschl2t+ρ20+ρ2∞. The correction term Φ0 is introduced into the control law so that the tracking error can converge to zero.Since the stability of the PI controller has been proven many times in early research, it will not be repeated here. Obviously, the boundedness ofε2t can be guaranteed via the PI control law. Furthermore, all the signals in the closed-loop are bounded. Thus, the prescribed transient performance and steady-state accuracy of the velocity subsystem are achieved.


## 3.1. Altitude Control Design

By considering equations (10), (11), and (17), there are x1≤ρ1tε1t+γd and xi≤ei+αi−1 with ei=xi−αi−1, where αi−1 is the virtual controller.Step 1.
It follows that fromLemma 1 and Young’s inequality, there is
(24)ε1tΛf1≤Λε1tx1ϒ1≤Λε1tϒ1ρ1tε1t+γd≤Λε1t2ϒ1ρ1t+Λϒ1ε1tγd2+14.
Choose the following Lyapunov function:(25)L1=12ε1t2.
The time derivative of (25) is as follows:
(26)L1≤ε1tΓ+Λf1+e2+α1−γ̇d≤ε1tΓ+Λe2+α1−γ̇d+ε1tΛf1≤ε1tΓ+Λe2+α1−γ̇d+ε1t2ρ1tΛϒ1+Λϒ1ε1tγd2+14=ε1tΓ+Λe2+α1−γ̇d+ϒ1ρ1tε1t+Λϒ12γd2ε1t+14.
Thus, choose the virtual control law as(27)α1=1Λ−k1ε1t−Γ−Λϒ1ρ1tε1t+Λϒ12γd2ε1t−γ̇d,where k1 is the design parameter. Substituting (27) into (26) results in
(28)L̇1≤−k1ε1t2+Λε1te2.Step 2.
Similar to Step1, there is
(29)e2f2=e2ϒ2x1+x2≤e2ϒ2ρ1tε1t+γd+e2+α1=ϒ2e22+e2ϒ2ρ1tε1t+γd+α1≤ϒ2e22+e2ϒ2ρ1tε1t+γd+α12+14.
Choose the following Lyapunov function:(30)L2=L1+12e22.
The time derivative of (30) is as follows:
(31)L̇2=L̇1+e2f2+x3−α̇1≤−k1ε1t2+ε1tΛe2+14+e2e3+x3−α̇1+ϒ2e22+e22ϒ2ρ1tε1t+γd+α12+14=−k1ε1t2+12+e2ε1tΛ+e3+α2−α̇1+ϒ2e2+e2ϒ2ρ1tε1t+γd+α12.
Thus, choose the virtual control law as(32)α2=−k2e2−ε1tΛ−ϒ2e2−e2ϒ2ρ1tε1t+γd+α12+α̇1,where k2 is the design parameter. Substituting (32) into (31) results in
(33)L̇2≤−k1ε1t2−k2e22+e2e3+12.Step 3.
Similar to Step2, there is
(34)e3f3≤ϒ3e32+e3ϒ3ρ1tε1t+γd+e2+α1+α22+14.
Choose the following Lyapunov function:(35)L3=L2+12e32.
The time derivative of (35) is as follows:
(36)L̇3=L̇2+e3f3+δe−α̇2≤−k1ε1t2−k2e22−k3e32+34+e3e3ϒ3ρ1tε1t+γd+e2+α1+α22+e2+ϒ3e3+δe−α̇2.
Thus, choose the control law as(37)δe=−k3e3−e2−ϒ3e3−e3ϒ3ρ1tε1t+γd+e2+α1+α22+α̇2,where k3 is the design parameter. It follows from (37) and (36) that (38) holds
(38)L̇3≤−k1ε1t2−k2e22+k3e32+34=−kL3+34,where k=min2k1,2k2,2k3. Then, solving the above differential equation, there is
(39)L3≤e−ktL30−34+34.Obviously, from Lemma2, transform error ε1t and tracking errors e2 and e3 are bounded. Furthermore, all the signals in the closed-loop are bounded. Thus, the transient performance and steady-state accuracy of the altitude subsystem are guaranteed.


## 3.2. Velocity Control Design

A PI controller is designed for the velocity subsystem because the velocity dynamics is much simpler than the altitude dynamic change.(40)Φ=−kV1ε2t−kV2∫0tε2tdt+Φ0,with ε2t=lnV~/ρ2t+1/1−V~/ρ2t, V~=V−Vref, and ρ2t=cschl2t+ρ20+ρ2∞. The correction term Φ0 is introduced into the control law so that the tracking error can converge to zero.Since the stability of the PI controller has been proven many times in early research, it will not be repeated here. Obviously, the boundedness ofε2t can be guaranteed via the PI control law. Furthermore, all the signals in the closed-loop are bounded. Thus, the prescribed transient performance and steady-state accuracy of the velocity subsystem are achieved.


## 4. Simulation Results

In order to verify the practicability of the designed controllers, this section will prove this via digital simulation experiments. The performance functions are selected asρht=csch0.1t+0.3+1.8, ρ1t=csch0.15t+2.8+0.05, and ρ2t=csch0.1t+0.2+0.9. The design parameters are chosen as follows: kh=8, k1=0.05, k2=0.1, k3=0.8, kV1=0.6, kV2=1, and Φ0=0.38. Meanwhile, the pneumatic parameters to account for uncertainties in the HFV dynamics are changed by defining C=C01+0.3sin0.1πt. The initial trim conditions of HFV are listed in Table 1. In the simulation process, when the time exceeds 30 s, the velocity subsystem and the altitude subsystem are interfered with dV=5sin0.1πt and dh=0.02sin0.5πt, respectively. In order to show the superiority of the proposed controller, the simulation results compare the prescribed performance tracking controller (PPT) with the novel robust control (NRC) in Ref. [39] and the concise neural control (CNC) in Ref. [40]. At the same time, the simulation experiments are carried out in the following two cases:Case 1.
The HFV velocity step and altitude step of cruise flight are required to be 100 m/s and 100 m, respectively.Case 2.
The HFV velocity input for cruise flight is required to be a signal with a step amplitude of 100 m/s per 100 seconds, and the height input is a square wave signal with an amplitude of 100 m and a period of 200 seconds.Table 1
HFV state initial value.
Item
Value
Units
V
2500
m/s
h
27000
m
γ
0
°
θ
1.5295
°
Q
0
°/s
η1
0.2857
—
η2
0.2350
—The simulation results are shown in Figures2 and 3. From the velocity and altitude tracking performance, three methods can realize stable tracking of reference inputs under the premise of interference. The PPT method has smaller overshoot and faster convergence speed. The altitude angle and elastic states of the three methods do not appear as high-frequency chattering phenomenon, but the PPT method has a smoother response. The control inputs of the three methods can be stable within the limited range, and the PPT method also has a smoother response. The transform error functions in both cases are bounded.Figure 2
Case1 simulation diagrams.
(a)
Velocity tracking performance
(b)
Altitude tracking performance
(c)
Altitude angle
(d)
Elastic states
(e)
Control inputs
(f)
Transform errorFigure 3
Case2 simulation diagrams.
(a)
Velocity tracking performance
(b)
Altitude tracking performance
(c)
Altitude angle
(d)
Elastic states
(e)
Control inputs
(f)
Transform error


## 5. Conclusions

In this paper, a prescribed performance tracking control is designed for HFV with model uncertainties. Based on the timescale principle, the HFV longitudinal motion model is decomposed into the velocity subsystem and the altitude subsystem. A novel performance function is designed which does not depend on the initial error. The performance function limits the tracking error within the prescribed range so that the system has satisfactory transient performance and steady-state accuracy. The control objective of the altitude subsystem is changed by model transformation, and the prescribed performance backstepping controller is designed. The prescribed performance PI controller is proposed for the velocity subsystem which has better engineering practicability. Obviously, the designed controllers are more convenient. Finally, the simulation results show that the proposed controller has good practicability. Meanwhile, compared with two other control methods, it has better transient performance and steady-state accuracy and a smoother response.



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*Source: 3505614-2019-10-13.xml*