# Second-Order Sliding Mode Disturbance Observer-Based Adaptive Fuzzy Tracking Control for Near-Space Vehicles with Prescribed Tracking Performance

Authors: Bin Fu; Kang Chen; Hang Guo
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

An adaptive fuzzy fault-tolerant tracking controller is developed for Near-Space Vehicles (NSVs) suffering from quickly varying uncertainties and actuator faults. For the purpose of estimating and compensating the mismatched external disturbances and modeling errors, a second-order sliding mode disturbance observer (SOSMDO) is constructed. By introducing the norm estimation approach, the negative effects of the quickly varying multiple matched disturbances can be handled. Meanwhile, a hierarchical fuzzy system (HFS) is employed to approximate and compensate the unknown nonlinearities. Several performance functions are introduced and the original system is transformed into one incorporating the desired performance criteria. Then, an adaptive fuzzy tracking control structure is established for the transformed system, and the predefined transient tracking performance can be guaranteed. The rigorous stability of the closed-loop system is proved by using the Lyapunov method. Finally, simulation results are presented to illustrate the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

As is well known, disturbances and uncertainties widely exist in industrial systems and may induce negative effects on control performance or even stability of practical control systems [1]. Therefore, a wealth of disturbance and uncertainty estimation and rejection methods, such as unknown input observer (UIO) [2, 3], perturbation observer [4], uncertainty and disturbance estimator [5], and disturbance observer (DO) [6], have been proposed. Meanwhile, a disturbance observer in frequency domain was developed in [7]. For the purpose of suppressing the periodic-disturbances, an adaptive periodic-disturbance observer is designed in [8]. In [9], a nonlinear disturbance observer is utilized in the robust control for spacecraft formation flying. In [10], a disturbance observer-based controller is developed for a magnetically suspended wheel with synchronous noise. In [11], a disturbance observer-based robust approaching control law is proposed for a tethered space robot. In [12, 13], several adaptive composite antidisturbance control methods have been investigated. In [14], a practical control method which can estimate the lumped disturbances consisting of both unknown uncertainties and external disturbances, called active disturbance rejection control (ADRC), was developed by Han and his collogues. Moreover, by using the equivalent-control-principle of the sliding mode control technique to estimate the lumped disturbances, sliding mode disturbance observer (SMDO) has been constructed [15]. In [16], to address the problem of disturbance rejection control for Markovian jump linear systems with matched and mismatched disturbances, an extended sliding mode observer based control law has been developed. In [17], sliding mode observers are utilized for state and disturbance estimation in electrohydraulic systems. For the vibration control of a train-car suspension with magnetorheological dampers, a SMDO-based controller has been investigated in [18]. For the air-breathing hypersonic flight vehicles subject to external disturbances and actuator saturations, a sliding mode exact disturbance observer (SMEDO) is exploited to exactly estimate the lumped disturbances [19, 20]. As is stressed in [21], SMDO possess strong robustness and accuracy to estimate a lumped disturbance including unknown external disturbances and parametric uncertainties. In [22], a sliding mode disturbance observer with switching-gain adaptation was proposed. In [23, 24], the second-order sliding mode disturbance observer, which can produce continuous estimation signals and possess strong robustness, had been proposed and applied to industrial systems.On the other hand, in the practical control systems, it is of significant importance to consider the transient control performances including the overshoot, undershoot, and convergence rate. However, there are few results focusing on this issue. Usually, the traditional adaptive control systems can only steer the tracking errors to converge to a residual set whose size resting with the control gains and the disturbances ranges [25, 26]. Recently, by utilizing appropriately defined functions to transform the original system into one that incorporates the desired performance criteria, an effective control scheme which can guarantee the prescribed transient performances has been proposed by Bechlioulis [27]. In [28], a fault-tolerant controller guaranteeing prescribed tracking performance has been proposed for a class of nonlinear uncertain systems. Moreover, the prescribed performance controllers for the cascade systems [29] and the nonaffine nonlinear large scale systems [30] have also been developed.Since NSVs are not constrained by orbital mechanics and fuel consumption, they can offer significant advantages to Low Earth Orbit (LEO) satellites and airplanes [31]. In general, the NSV has larger flight envelope, rapid flight speed, and time-varying aerodynamic characteristics; the controller design becomes challenging and it is necessary to develop effective control methods to guarantee the safety and reliability [32]. In the past decades, a number of advanced control approaches have been developed for the NSVs. In [33], by introducing the neural networks into an adaptive backstepping controller, an effective attitude controller has been proposed. For the NSV suffering from the dynamical uncertainties, an adaptive functional link network control structure has been constructed in [34]. In [35], a robust attitude controller has been designed for NSVs subjected to time-varying disturbances. In [36], a globally convergent Levenberg–Marquardt (LM) algorithm based on Takagi–Sugeno fuzzy training has been proposed for the NSV. In [37], a terminal sliding mode controller combined with dynamic sliding mode was designed based on nonlinear disturbance observer.In spite of the progress, the control results for the NSVs with guaranteed transient performances have rarely been reported. Moreover, in practice, the aerodynamic parameters perturbations, measurement errors, actuator faults, and wind effects may induce significant uncertainties in the flight control systems of NSVs. To the best of the authors’ knowledge, the existing controllers are commonly designed for the NSVs suffering from only constant disturbances, and the aforementioned quickly varying multiple uncertainties cannot be handled. Therefore, in this paper, we are dedicated to designing an adaptive tracking control structure which can guarantee the prescribed transient tracking performance and possess satisfactory disturbance attention ability. By introducing a second-order sliding mode disturbance observer, the time-varying mismatched uncertainties can be estimated and compensated. With the aid of the hierarchical fuzzy approximator and a novel nonlinear function, the negative effects of the quickly varying multiple matched disturbances can be handled. Based on the performance functions and error transformation, the robustness and predefined tracking performance can be ensured. The adverse actuator derivations are also overcome by the proposed controller. Compared with the previous works, the contributions of this paper are summarized as follows.(1)
An effective adaptive fuzzy control algorithm, which can guarantee the prescribed transient tracking performances including the convergence rate, the tracking error, and the overshoot, is developed for the NSVs.(2)
The NSV model suffering from the multisource quickly varying uncertainties, including the actuator faults, the wind effects, aerodynamic uncertainties, and measurement errors, is established.(3)
The adverse effects of the quickly varying uncertainties can be circumvented. The theoretic developments of this paper are valuable for handling the time-varying disturbances.


## 2. Problem Formulation and Preliminaries

### 2.1. NSV Model

According to [38, 39], the attitude dynamic model of NSVs can be described by(1)α˙=ωz+ωysin⁡αtan⁡β-ωxcos⁡αtan⁡β-Y+mgcos⁡θcos⁡γvmVcos⁡ββ˙=ωycos⁡α+ωxsin⁡α+Z-mgcos⁡θsin⁡γvmVγ˙v=-ωysin⁡αcos⁡β+ωxcos⁡αcos⁡β+Ysin⁡θsin⁡γv+tan⁡β+Zsin⁡θcos⁡γv+mgcos⁡βcos⁡γvtan⁡βmVω˙x=r1ωxωz+r2ωyωz+r3Mx+r4Myω˙y=q1ωxωz+q2ωyωz+r4Mx+q3Myω˙z=p1ωxωy+p2ωx2-ωy2+p3Mzwhere α,β, and γv are the angle of attack, sideslip angle, and bank angle of the NSV, respectively. θ is the flight path angle. Y and Z represent lift and lateral force. V is the velocity of the vehicle. ωx,ωy, and ωz are the roll, pitch, and yaw angular rates of the vehicle, respectively. The inertial parameters p1,p2,p3,q1,q2,q3,r1,r2,r3,r4 can be formulated by(2)p1=Jx-JyJz,p2=JxyJz,p3=1Jz,q1=JxJz-Jxy2-Jx2JxJy-Jxy2,q2=-JxyJx-Jy+JzJxJy-Jxy2,q3=JxJxJy-Jxy2,r1=-JxyJx+Jy-JzJxJy-Jxy2,r2=Jy2-Jxy2-JyJzJxJy-Jxy2,r3=JyJxJy-Jxy2,r4=JxyJxJy-Jxy2The aerodynamic forces can be described as(3)Y=CYqSZ=CZqSwhere(4)CY=CYαα+CYδzδz+CYδyδyCZ=CZββ+CZδyδy+CZδxδxThe aerodynamic moment are as follows: (5)Mx=qSLmx0+mxαα+mxββ+mxδyδy+mxδxδx+mxω-xω-xMy=qSLmy0+myαα+myββ+myδyδy+myδxδx+myω-yω-yMz=qSLmz0+mzαα+mzββ+mzδzδz+mzω-zω-zwhere mx0,my0,mz0 denote the aerodynamic coefficients when the deflection angles are zero. mxω-x,myω-y,myω-y represent the coefficients of aerodynamic damping torque. CY is the aerodynamic coefficient of the lift force; CZ is the lateral force coefficient. The aerodynamic partial derivatives are represented by mxα,mxβ,myα,myβ,mzα,mzβ,mxδx,mxδy,myδx,myδy,mzδz. δx,δy,δz are the deflection angles of control surfaces. S is the reference area. L is the reference length. q is dynamic pressure. ω-x,ω-y,ω-z are calculated as (6)ω-x=LωxV,ω-y=LωyV,ω-z=LωzVIt should be highlighted that multiple uncertainties including the aerodynamic uncertainties, measurement errors, and unmodeled dynamics are often encountered in the NSV. Meanwhile, the NSVs often experience a complex flight environment, and the undesired stochastic winds have to be considered. LetΔα and Δβ represent the additional angle of attack and the additional sideslip angle, respectively. Then the disturbance forces and torques caused by the unknown winds can be described as (7)ΔYWind=qSCYαΔα,ΔZWind=qSCZβΔβΔMxWind=qSLmxαΔα+mxβΔβΔMyWind=qSLmyαΔα+myβΔβΔMzWind=qSLmzαΔα+mzβΔβMoreover, the aerodynamic-perturbation-caused disturbance forces and torques are defined as ΔYPtb,ΔZPtb,ΔMxPtb,ΔMyPtb,ΔMzPtb.In general, the NSVs are commonly equipped with the electromechanical aerodynamic rudders. However, there often exist multifarious faults in the aerodynamic rudders. As is revealed in [3, 4], actuator faults constitute the main reason for the undesired control performance and instability of the flight control system. Define u=δzδyδxT. The faulty actuator model can be provided as (8)uc=u+Δδwhere u and uc denote the input and output of the actuator, respectively. Δδ denotes the deviation caused by the actuator faults. It is supposed that Δδ≤Δδ-.The neglect of the above-mentioned multiple uncertainties and actuator faults may cause worse control performance and instability of the flight control system. The intrinsic high dynamic characteristics and the strong coupled properties make it a challenge to establish an effective control scheme. Definex1=αβγvT and x2=ωzωyωxT. Taking the multiple uncertainties into consideration, we can get the following NSV model:(9)x˙1=Ax2+f1x1,x2+Δf1x1,x2+DFx˙2=f2x1,x2+Δf2x1,x2+Bu+BΔδ+DMwhere(10)A=1sin⁡αtan⁡β-cos⁡αtan⁡β0cos⁡αsin⁡α0-sin⁡αcos⁡βcos⁡αcos⁡βB=qSLp3000q3r40r4r3mzδz000myδymyδx0mxδymxδxf1x1,x2=-Y+mgcos⁡θcos⁡γvmVcos⁡βZ-mgcos⁡θsin⁡γvmVYsin⁡θsin⁡γv+tan⁡β+Zsin⁡θcos⁡γv+mgcos⁡θcos⁡γvtan⁡βmVf2x1,x2=p1ωxωy+p2ωx2-ωy2q1ωxωz+q2ωyωzr1ωxωz+r2ωyωz+qSLp3000q3r40r4r3mz0my0mx0+p3000q3r40r4r3Cmqω-zCnrω-yClpω-xDF=DF,0+DF,1DF,0=-ΔYmVcos⁡βΔZmVΔYsin⁡θsin⁡γv+tan⁡β+ΔZsin⁡θcos⁡γvmVDM=qSLp3000q3r40r4r3ΔMxPtb+ΔMxWind+ΔMxAss+ΔMxFabΔMyPtb+ΔMyWind+ΔMyAss+ΔMyFabΔMzPtb+ΔMzWind+ΔMzAss+ΔMzFab+ΔBuΔY=ΔYPtb+ΔYWindΔZ=ΔZPtb+ΔZWindΔf1(x1,x2) and Δf2(x1,x2) are the unknown nonlinearities caused by the model simplification. DF,1 denotes the disturbances caused by inaccurate measurement information. ΔB represents the uncertainty part of the control distribution matrix, which is given as (11)ΔB=qSLp3000q3r40r4r3Δmzδz000ΔmyδyΔmyδx0ΔmxδyΔmxδxIn practice, the transient performance including the convergence rate, the tracking error, and the predefined maximum overshoot should be guaranteed. Therefore, the design objective of this work is to establish an effective control structure to force the angle of attackα, the sideslip angle β, and the bank angle γV to track the reference trajectories αr,βr,γV,r with predefined transient performance in the simultaneous presence of the actuator faults and multiple uncertainties.


### 2.2. Assumptions and Supporting Lemmas

The following assumptions and lemmas are necessary in this work.Assumption 1.
The actuator deviations are supposed to satisfyΔδ≤Δδ-.Assumption 2.
The disturbances existing in the NSV system are assumed to be bounded; i.e.,Δf1(x1,x2)≤d-f,DF≤d-DF,DM≤d-DM.Lemma 3.
Consider the system(12)x˙=fx,f0=0,x∈Rn,x0=x0where f:D→Rn is continuous on an open neighborhood D and the origin is 0. Suppose there is a continuous function Vx:D→Rn defined on U⊆D with the origin 0 such that the following conditions hold:
1   V x is positive definite on D⊆Rn.
( 2 ) There exist real numbers k>0 and υ∈0,1, such that V˙x+kVυx≤0, and then, system (12) is locally finite-time stable. The settling time, depending on the initial state x0=x0, satisfies(13)Tx0≤Vx01-υk1-υfor all x0 in some open neighborhood of the origin. If D=Rn and Vx is also unbounded, system (12) is globally finite-time stable.Lemma 4 (see [26]).
Given anyε>0 and z∈Rn, then one has(14)0≤z-zTzzTz+ε2<ε


### 2.3. Hierarchical Fuzzy Systems

For the purpose of compensating the unknown nonlinear functions, we introduce a HFS in the controller. Commonly, the HFS contains several low dimensional fuzzy systems. The structure of the HFS is provided in Figure1. Let x=x1,x2,⋯,xnT represent the input of the HFS. Then the first subsystem can be established by using the following rules.Figure 1
The structure of the hierarchical fuzzy system.Rule l. If x1 is F1l and x2 is F2l, then y1 is G1l,l=1,2,...,N1.N 1 is the total number of fuzzy rules for the first fuzzy subsystem; F1l,F2l,G1l are the corresponding fuzzy sets ,μFklxk,k=1,2, and μGly1 represent the fuzzy functions. A subsystem includes a singleton fuzzifier, a center-average defuzzifier, and a product inference engine. The following equation provides the output of a fuzzy subsystem:(15)y1x1,x2=∑l=1N1w1,lμF1lx1μF2lx2∑l=1N1μF1lx1μF2lx2=W1TΦ1x1,x2where W1=w1,1,⋯,w1,N1T. (16)Φ1x1,x2=Φ1,1x1,x2,⋯,Φ1,N1x1,x2TΦ1,lx1,x2=μF1lx1μF2lx2∑l=1N1μF1lx1μF2lx2Then, we can establish thei-th (i=2,⋯,n-1 ) fuzzy subsystem with the aid of the following rules.Rule l. If xi+1 is Fi+1l and yi-1 is Gi-1l, then yi is Gil,l=1,2,...,Ni.x i + 1 and yi-1 are the inputs, and yi is the output. Hence, yi, which is the output of thei-th fuzzy subsystem, is calculated as (17)yi=∑l=1Niwi,lμGi-1lyi-1μFi+1lxi+1∑l=1NiμGi-1lyi-1μFi+1lxi+1=WiTΦiyi-1,xi+1Finally, a hierarchical fuzzy system can be established and the final output can be given as(18)yn-1=Wn-1TΦn-1yn-2,xnIt is clear that the total number of fuzzy rules is ∏i=1nNi2.Lemma 5 (see [40]).
DefineU=[a1,b1]×⋯×[an,bn]. Consider a nonlinear function fx1,⋯,xn, which is continuously differentiable and defined on U. Then we can construct a hierarchical fuzzy system yx1,⋯,xn within (15) ~ (18) such that(19)fx-yx∞≤∑i=1n∂f∂xi∞+∂y∂xi∞b-aNi-1where x=x1,⋯,xnT.Lemma 6 (see [40]).
DefineU=[a1,b1]×⋯×[an,bn]. Consider any function fx1,⋯,xn which is continuously differentiable and defined on U. Then one can construct a hierarchical fuzzy system yx1,⋯,xn within (15) ~ (18) such that supx∈U⁡f(x)-y(x)<ε, where ε is an arbitrarily small positive constant.


## 2.1. NSV Model

According to [38, 39], the attitude dynamic model of NSVs can be described by(1)α˙=ωz+ωysin⁡αtan⁡β-ωxcos⁡αtan⁡β-Y+mgcos⁡θcos⁡γvmVcos⁡ββ˙=ωycos⁡α+ωxsin⁡α+Z-mgcos⁡θsin⁡γvmVγ˙v=-ωysin⁡αcos⁡β+ωxcos⁡αcos⁡β+Ysin⁡θsin⁡γv+tan⁡β+Zsin⁡θcos⁡γv+mgcos⁡βcos⁡γvtan⁡βmVω˙x=r1ωxωz+r2ωyωz+r3Mx+r4Myω˙y=q1ωxωz+q2ωyωz+r4Mx+q3Myω˙z=p1ωxωy+p2ωx2-ωy2+p3Mzwhere α,β, and γv are the angle of attack, sideslip angle, and bank angle of the NSV, respectively. θ is the flight path angle. Y and Z represent lift and lateral force. V is the velocity of the vehicle. ωx,ωy, and ωz are the roll, pitch, and yaw angular rates of the vehicle, respectively. The inertial parameters p1,p2,p3,q1,q2,q3,r1,r2,r3,r4 can be formulated by(2)p1=Jx-JyJz,p2=JxyJz,p3=1Jz,q1=JxJz-Jxy2-Jx2JxJy-Jxy2,q2=-JxyJx-Jy+JzJxJy-Jxy2,q3=JxJxJy-Jxy2,r1=-JxyJx+Jy-JzJxJy-Jxy2,r2=Jy2-Jxy2-JyJzJxJy-Jxy2,r3=JyJxJy-Jxy2,r4=JxyJxJy-Jxy2The aerodynamic forces can be described as(3)Y=CYqSZ=CZqSwhere(4)CY=CYαα+CYδzδz+CYδyδyCZ=CZββ+CZδyδy+CZδxδxThe aerodynamic moment are as follows: (5)Mx=qSLmx0+mxαα+mxββ+mxδyδy+mxδxδx+mxω-xω-xMy=qSLmy0+myαα+myββ+myδyδy+myδxδx+myω-yω-yMz=qSLmz0+mzαα+mzββ+mzδzδz+mzω-zω-zwhere mx0,my0,mz0 denote the aerodynamic coefficients when the deflection angles are zero. mxω-x,myω-y,myω-y represent the coefficients of aerodynamic damping torque. CY is the aerodynamic coefficient of the lift force; CZ is the lateral force coefficient. The aerodynamic partial derivatives are represented by mxα,mxβ,myα,myβ,mzα,mzβ,mxδx,mxδy,myδx,myδy,mzδz. δx,δy,δz are the deflection angles of control surfaces. S is the reference area. L is the reference length. q is dynamic pressure. ω-x,ω-y,ω-z are calculated as (6)ω-x=LωxV,ω-y=LωyV,ω-z=LωzVIt should be highlighted that multiple uncertainties including the aerodynamic uncertainties, measurement errors, and unmodeled dynamics are often encountered in the NSV. Meanwhile, the NSVs often experience a complex flight environment, and the undesired stochastic winds have to be considered. LetΔα and Δβ represent the additional angle of attack and the additional sideslip angle, respectively. Then the disturbance forces and torques caused by the unknown winds can be described as (7)ΔYWind=qSCYαΔα,ΔZWind=qSCZβΔβΔMxWind=qSLmxαΔα+mxβΔβΔMyWind=qSLmyαΔα+myβΔβΔMzWind=qSLmzαΔα+mzβΔβMoreover, the aerodynamic-perturbation-caused disturbance forces and torques are defined as ΔYPtb,ΔZPtb,ΔMxPtb,ΔMyPtb,ΔMzPtb.In general, the NSVs are commonly equipped with the electromechanical aerodynamic rudders. However, there often exist multifarious faults in the aerodynamic rudders. As is revealed in [3, 4], actuator faults constitute the main reason for the undesired control performance and instability of the flight control system. Define u=δzδyδxT. The faulty actuator model can be provided as (8)uc=u+Δδwhere u and uc denote the input and output of the actuator, respectively. Δδ denotes the deviation caused by the actuator faults. It is supposed that Δδ≤Δδ-.The neglect of the above-mentioned multiple uncertainties and actuator faults may cause worse control performance and instability of the flight control system. The intrinsic high dynamic characteristics and the strong coupled properties make it a challenge to establish an effective control scheme. Definex1=αβγvT and x2=ωzωyωxT. Taking the multiple uncertainties into consideration, we can get the following NSV model:(9)x˙1=Ax2+f1x1,x2+Δf1x1,x2+DFx˙2=f2x1,x2+Δf2x1,x2+Bu+BΔδ+DMwhere(10)A=1sin⁡αtan⁡β-cos⁡αtan⁡β0cos⁡αsin⁡α0-sin⁡αcos⁡βcos⁡αcos⁡βB=qSLp3000q3r40r4r3mzδz000myδymyδx0mxδymxδxf1x1,x2=-Y+mgcos⁡θcos⁡γvmVcos⁡βZ-mgcos⁡θsin⁡γvmVYsin⁡θsin⁡γv+tan⁡β+Zsin⁡θcos⁡γv+mgcos⁡θcos⁡γvtan⁡βmVf2x1,x2=p1ωxωy+p2ωx2-ωy2q1ωxωz+q2ωyωzr1ωxωz+r2ωyωz+qSLp3000q3r40r4r3mz0my0mx0+p3000q3r40r4r3Cmqω-zCnrω-yClpω-xDF=DF,0+DF,1DF,0=-ΔYmVcos⁡βΔZmVΔYsin⁡θsin⁡γv+tan⁡β+ΔZsin⁡θcos⁡γvmVDM=qSLp3000q3r40r4r3ΔMxPtb+ΔMxWind+ΔMxAss+ΔMxFabΔMyPtb+ΔMyWind+ΔMyAss+ΔMyFabΔMzPtb+ΔMzWind+ΔMzAss+ΔMzFab+ΔBuΔY=ΔYPtb+ΔYWindΔZ=ΔZPtb+ΔZWindΔf1(x1,x2) and Δf2(x1,x2) are the unknown nonlinearities caused by the model simplification. DF,1 denotes the disturbances caused by inaccurate measurement information. ΔB represents the uncertainty part of the control distribution matrix, which is given as (11)ΔB=qSLp3000q3r40r4r3Δmzδz000ΔmyδyΔmyδx0ΔmxδyΔmxδxIn practice, the transient performance including the convergence rate, the tracking error, and the predefined maximum overshoot should be guaranteed. Therefore, the design objective of this work is to establish an effective control structure to force the angle of attackα, the sideslip angle β, and the bank angle γV to track the reference trajectories αr,βr,γV,r with predefined transient performance in the simultaneous presence of the actuator faults and multiple uncertainties.


## 2.2. Assumptions and Supporting Lemmas

The following assumptions and lemmas are necessary in this work.Assumption 1.
The actuator deviations are supposed to satisfyΔδ≤Δδ-.Assumption 2.
The disturbances existing in the NSV system are assumed to be bounded; i.e.,Δf1(x1,x2)≤d-f,DF≤d-DF,DM≤d-DM.Lemma 3.
Consider the system(12)x˙=fx,f0=0,x∈Rn,x0=x0where f:D→Rn is continuous on an open neighborhood D and the origin is 0. Suppose there is a continuous function Vx:D→Rn defined on U⊆D with the origin 0 such that the following conditions hold:
1   V x is positive definite on D⊆Rn.
( 2 ) There exist real numbers k>0 and υ∈0,1, such that V˙x+kVυx≤0, and then, system (12) is locally finite-time stable. The settling time, depending on the initial state x0=x0, satisfies(13)Tx0≤Vx01-υk1-υfor all x0 in some open neighborhood of the origin. If D=Rn and Vx is also unbounded, system (12) is globally finite-time stable.Lemma 4 (see [26]).
Given anyε>0 and z∈Rn, then one has(14)0≤z-zTzzTz+ε2<ε


## 2.3. Hierarchical Fuzzy Systems

For the purpose of compensating the unknown nonlinear functions, we introduce a HFS in the controller. Commonly, the HFS contains several low dimensional fuzzy systems. The structure of the HFS is provided in Figure1. Let x=x1,x2,⋯,xnT represent the input of the HFS. Then the first subsystem can be established by using the following rules.Figure 1
The structure of the hierarchical fuzzy system.Rule l. If x1 is F1l and x2 is F2l, then y1 is G1l,l=1,2,...,N1.N 1 is the total number of fuzzy rules for the first fuzzy subsystem; F1l,F2l,G1l are the corresponding fuzzy sets ,μFklxk,k=1,2, and μGly1 represent the fuzzy functions. A subsystem includes a singleton fuzzifier, a center-average defuzzifier, and a product inference engine. The following equation provides the output of a fuzzy subsystem:(15)y1x1,x2=∑l=1N1w1,lμF1lx1μF2lx2∑l=1N1μF1lx1μF2lx2=W1TΦ1x1,x2where W1=w1,1,⋯,w1,N1T. (16)Φ1x1,x2=Φ1,1x1,x2,⋯,Φ1,N1x1,x2TΦ1,lx1,x2=μF1lx1μF2lx2∑l=1N1μF1lx1μF2lx2Then, we can establish thei-th (i=2,⋯,n-1 ) fuzzy subsystem with the aid of the following rules.Rule l. If xi+1 is Fi+1l and yi-1 is Gi-1l, then yi is Gil,l=1,2,...,Ni.x i + 1 and yi-1 are the inputs, and yi is the output. Hence, yi, which is the output of thei-th fuzzy subsystem, is calculated as (17)yi=∑l=1Niwi,lμGi-1lyi-1μFi+1lxi+1∑l=1NiμGi-1lyi-1μFi+1lxi+1=WiTΦiyi-1,xi+1Finally, a hierarchical fuzzy system can be established and the final output can be given as(18)yn-1=Wn-1TΦn-1yn-2,xnIt is clear that the total number of fuzzy rules is ∏i=1nNi2.Lemma 5 (see [40]).
DefineU=[a1,b1]×⋯×[an,bn]. Consider a nonlinear function fx1,⋯,xn, which is continuously differentiable and defined on U. Then we can construct a hierarchical fuzzy system yx1,⋯,xn within (15) ~ (18) such that(19)fx-yx∞≤∑i=1n∂f∂xi∞+∂y∂xi∞b-aNi-1where x=x1,⋯,xnT.Lemma 6 (see [40]).
DefineU=[a1,b1]×⋯×[an,bn]. Consider any function fx1,⋯,xn which is continuously differentiable and defined on U. Then one can construct a hierarchical fuzzy system yx1,⋯,xn within (15) ~ (18) such that supx∈U⁡f(x)-y(x)<ε, where ε is an arbitrarily small positive constant.


## 3. Main Results

### 3.1. Performance Functions

Defineeα=α-αr,eβ=β-βr and eγV=γV-γV,r. Then, for the purpose of guaranteeing the predefined control performance, we introduce the performance functions in the following text. Define the following exponentially decaying functions pit,i∈α,β,γV as the performance functions:(20)pit=pi,0-pi,∞exp⁡-ait+pi,∞,i∈α,β,γVwhere pi,0>pi,∞>0 and ai>0. Define b_i,b-i>0. If the inequality -b_ipi(t)<ei(t)<b-ipi(t) holds for t≥0, then the transient performance is ensured. To make this point more clear, we make the following explanations. b-ipi,0 and -b_ipi,0 restrict the range of the overshoot and the lower bound of the undershoot of eit. The convergence speed of eit is restrained by the decreasing speed of pit. The steady error eit is constrained by max⁡{b_ipi,∞,b-ipi,∞}.Consider the following strictly increasing function:(21)si=Teipi=ln⁡b_i+eipi-ln⁡b-i-eipi,i∈α,β,γVClearly, function (21) satisfies(22)limei/pi→-b_i⁡Teipi=-∞,limei/pi→b-iTeipi=+∞It is not difficult to verify that the inverse function S-1∙ is well-defined and strictly increasing as well. Obviously, if si is bounded, eit/pit remains within a compact subset of -b_i,b-i. As a result, the control issue forces si to be bounded.


### 3.2. Second-Order Sliding Mode Disturbance Observer

Definex1,r=αr,βr,γV,rT and e1=x1-x1,r. Consider the NSV model given in (9); we can get the following dynamic equation: (23)e˙1=Ax2+f1x1,x2+Δf1x1,x2+DF-x1,rx˙2=f2x1,x2+Δf2x1,x2+Bu+BΔδ+DMUsing the strictly increasing function provided in (21), it can be proven that (24)s˙=s-Ax2+f1x1,x2+Δf1x1,x2+DF-x1,r-ψx˙2=f2x1,x2+Δf2x1,x2+Bu+BΔδ+DMwhere (25)s=sα,sβ,sγVT,s-=diag⁡s-α,s-β,s-γVψ=ψα,ψβ,ψγVTs-i=∂T∂ei/pi1pi,ψi=eip˙ipi,i∈α,β,γVConsider the mismatched uncertainties dmis=Δf1(x1,x2)+DF existing in system, a second-order sliding mode disturbance observer is introduced for estimation and compensation. The SOSMDO is designed as (26)d^mis=v,v=-k1ξξ1/2+z-k2ξ,z˙=-k3ξξ-k4ξξ=z0-e1,z˙0=Ax2+f1x1,x2-x1,r+vIn this work, we divide the mismatched uncertainties into two parts. One is the unknown disturbance related to the system states; the other is the time-varying unknown uncertainty. In other words, (27)dmis=d1t,ξ+∫0td2τdτWithout loss of generality, it is supposed that there exist constants g1,g2 such that (28)d1t,ξ≤g1ξ,d2≤g2Theorem 7.
Consider system (23). If one selects k1,⋯,k4 such that(29)k1>2g2,k2>2g1,k3>max⁡k-3Θ,k-3Υ,k4>max⁡k-4Θ,k-4Υ,where (30)k-3Θ=3g2+2g22k12,k-4Θ=3/2k12k2+3g2k22k3k12-2g22-3g2k12+k22+32k2g1k-3Υ=9/16k1g12k2k2-2g1+1/2k12g1-2k12k2+k2g2k2-2g1k-4Υ=1k2-2g19/16k1g12k2+1/2g12/k22k2k3+2k12-g2-2k3+1/2k12g1-9/16k1g12+2k22g1+1/4k2g12k2-2g1 then the estimation error of the mismatched uncertainties can be forced to zero in finite time.Proof.
From (23) and (26), we can get the following closed-loop dynamic equation:(31)ξ˙=-k1ξξ1/2+z-k2ξ-dmisz˙=-k3ξξ-k4ξ Consider the following Lyapunov function:(32)Vξ,z=2k3ξ+k4ξTξ+12zTz+ηTηwhere η=k1ξξ-1/2+k2ξ-z. Taking the derivatives of (32) yields(33)V˙ξ,z=2k3+k122ξTξ˙ξ+2k122+k4ξTξ˙+2zTz˙+32k1k2ξTξ˙ξ1/2-k2ξ˙Tz+ξTz˙-k1-12ξTξ˙zTξξ5/2+z˙TξzTξ˙ξ1/2 Substituting (31) into (33) yields(34)V˙ξ,z=-k1k3+k132ξ2ξ3/2+32k1k2ξTd1ξ1/2-k2k4+k23ξ2-k4k1+52k1k22ξ2ξ1/2+k12ξTzξ+2k22ξTz+3k1k2ξTzξ1/2-k2z2+k12ξTzzTξξ5/2-k1zTzξ1/2+2k3+k122ξTd1ξ+2k4+k22ξTd1-k3k2+2k12k2ξ2ξ-k2d1Tz+k12ξTd1zTξξ5/2-k1zTd1ξ1/2+2zTd2-k2ξTd2-k1d2Tξξ1/2 Furthermore, (35)V˙ξ,z≤-k1k3+k132ξ1/2-k3k2+2k12k2ξ-k2k4+k23ξ2-k4k1+52k1k22ξ3/2+k12ξTzξ+2k22ξTz+3k1k2ξTzξ1/2-k2z2+k12ξTz2ξ5/2+2k3+k122ξTd1ξ+2k4+k22ξTd1+32k1k2ξTd1ξ1/2+k2d1Tz+k12ξTd1zTξξ5/2+k1zTd1ξ1/2+2zTd2+k2ξTd2+k1d2Tξξ1/2By using (28) we can get that (36)V˙ξ,z≤-k1k3+k132ξ1/2-k3k2+2k12k2ξ-k4k1+52k1k22ξ3/2+k12z-k2k4+k23ξ2+2k22ξz+3k1k2ξ1/2z-k2z2+k12z2ξ1/2+2k3+k122g1ξ+2k4+k22g1ξ2+32k1k2ξ3/2g1+k2g1ξz+32k1g1ξ1/2z+2g2z+k2g2ξ+k1g2ξ1/2Denote ζ=colξ1/2,ξ,z; then we can rewrite (36) as(37)V˙≤-1ξ1/2ζTΘζ-ζTΥζwhere(38)Θ=Θ110Θ130Θ22Θ23Θ31Θ32Θ33,Υ=Υ110Υ130Υ22Υ23Υ31Υ32Υ33,Θ11=12k13+k1k3-g2k1,Θ13=-12k12-g2,Θ22=k4k1+52k22k1-32k1k2g1,Θ23=-32k1k2,Θ31=Θ13,Θ32=Θ23,Θ33=12k1,Υ11=k2k3+2k12k2-k2g2-2k3+12k12g1,Υ13=-34k1g1,Υ22=k4k2+k23-k22+2k4g1,Υ23=-k22-12k2g1,Υ31=Υ13,Υ32=Υ23,Υ33=k2Clearly, by using condition (29),we can easily get that Θ>0 and Υ>0. Then, it follows from Rayleigh’s inequality that(39)V˙≤-1ξ1/2ζTΘζ≤-1ξ1/2λminΘζ2By defining ζ-=colξξ-1/2,ξ,z, we can rewrite (39) as(40)V˙≤-1ξ1/2λminΘζ-2Furthermore, (41)V˙≤-1ξ1/2λminΘλmaxΥVSince V1/2>λminΥξ1/2, it can be proven that (42)V˙≤-λminΘλminΥλmaxΥV1/2According to Lemma 3, it can be observed that the equilibrium point ξ,z=0 can be reached in finite time. Then by substituting ξ=z=0 into (31), we know ξ˙=0. As a result, ξ=ξ˙=0 is achieved in finite time. Define the estimation error of the mismatched uncertainty as d~mis=d^mis-dmis. Then from (26), it can be proven that ξ˙=d~mis. Therefore, it can be concluded that d~mis is forced to zero in finite time. The proof is completed.


### 3.3. Hierarchical Fuzzy Approximation-Based Adaptive Fault-Tolerant Tracking Control

Considering the following transformed dynamic equation(43)s˙=s-Ax2+f1x1,x2+Δf1x1,x2+DF-x1,r-ψx˙2=f2x1,x2+Δf2x1,x2+Bu+BΔδ+DMwe design the virtual control signal as follows: (44)x2c=s-A-1-K1s-s-f1x1,x2-s-d^mis+s-x˙1,r+s-ψwhere K1∈R3×3. To avoid the excessive complexity of control design, a first-order filter is established as(45)τx˙2d+x2d=x2c-τATs-Ts,x2d0=x2c0where τ∈R3×3 is the parameter matrix of the filter. Denote e2=x2-x2d,y=x2d-x2c. Then from (43) ~ (45) we can infer the closed-loop dynamic equation as(46)s˙=s-Ae2+y-K1s-s-d~misy˙=-1τy-ATs-Ts-x˙2cAccordingly, along (43), take the derivate of e2 as(47)e˙2=f2x1,x2+Δf2x1,x2+Bu+BΔδ+DM-x˙2dIn view of the unknown nonlinearities existing in (47), a hierarchical fuzzy logic system WTΦx is constructed for compensation. Ideally, Δf2x1,x2=WTΦx+εϕ. W is the weight matrix of the HFLS, N is the number of the rules, and Φx∈RN is the membership function. The final control law is developed as(48)u=B-1-K2e2-f2x1,x2-W^TΦx+x˙2d-ATs-Ts-D-^Me2e2Te2+εD2where K2=diag⁡k21,k22,k23; W^∈RN×3 is the estimation of W. D-^M represent the estimations of D-M, D-M=supt≥0DM+BΔδ+εϕ. The adaptive laws are given by (49)D-^˙M=ηDe2Te2e2Te2+εD2-σDD-^MW^˙=ΓΦxe2T-σWW^σD,ηD,σW>0 are design constants. Γ>0 are the gain matrixes of the adaptive laws. The structure of the proposed control method is provided in Figure 2.Figure 2
The structure of the proposed control method.Theorem 8.
Consider system (43). Suppose the estimation error of the mismatched uncertainty is bounded. Then, by using control laws (44) and (48) and adaptive laws (49), it can be ensured that all signals of the overall closed-loop system are globally uniformly bounded. As a result, the predefined control performance bounds can be guaranteed.Proof.
Consider the following Lyapunov function candidate:(50)V=V1+V2V1=12sTs+12yTyV2=12e2Te2+12ηDD-~MD-~M+12trW~TΓ-1W~Then, along (46), the derivative of V1 can be taken as(51)V˙1=sTs-Ae2+y-sTK1s-sTs-d~mis-yTyτ-yTATs-Ts-yTx˙2cResorting to Young’s inequality, it can be proven that (52)-yTx˙2c≤yTy2τ+τεc2-sTs-d~mis≤14sTs+ε-mis2where ε-mis=supt≥0s-d~mis,εc=supt≥0x˙2c. Therefore, we can rewrite (51) as(53)V˙1=sTs-Ae2-sTK1-14Is-yTy2τ+ε1where (54)ε1=ε-mis2+τεc2 From (47) and (48), it can be obtained that (55)e˙2=-K2e2-D-^Me2e2Te2+εD2-W~TΦx-ATs-Ts+BΔδ+DM+εϕwhere G~=G^-G, D~ω=D-^ω-D-ω. Take the derivative of V2 along (55) as(56)V˙2=-e2TK2e2-e2TATs-Ts-e2TD-^ωe2e2Te2+εD2-e2TW~TΦx+e2TBΔδ+DM+εϕ+1ηDD-~MD-^˙M+trW~TΓ-1W^˙By using Lemma 4, we can get that (57)e2TBΔδ+DM+εϕ≤e2D-M≤D-Me2Te2e2Te2+εD2+D-MεDFurther(58)-e2TD-^Me2e2Te2+εD2+e2TBΔδ+DM+εϕ≤-D-~Me2Te2e2Te2+εD2+D-MεDCombining (56) ~ (58) yields (59)V˙2≤-e2TK2e2-e2TATs-Ts-D-~ωe2Te2e2Te2+εD2-e2TW~TΦx+1ηDD-~MD-^˙M+trW~TΓ-1W^˙+D-MεDBy using the vector trace identity aTb=trbaT we can obtain that (60)V˙2≤-e2TK2e2-e2TATs-Ts-D-~ωe2Te2e2Te2+εD2-e2TW~2TΦ2x+1ηDD-~ωD-^˙ω+trW~TΓ-1W^-Φxe2T+D-MεDSubsisting (49) into (60), we come to(61)V˙2≤-e2TK2e2-e2TATs-Ts-σDD-~MD-^M-σWtrW~TW^+D-MεDThen it follows from (53) and (61) that (62)V˙≤-e1TK1-14Ie1-e2TK2e2-yTy2τ-σWtrW~TW^-σDD-~MD-^M+ε1+D-MεDClearly, (63)-W~TW^≤-12W~TW~+12WTW,-D-~ωD-^ω≤-12D-~ω2+12D-ω2Therefore, it can be checked that (64)V˙≤-e1TK1-14Ie1-e2TK2e2-yTy2τ-12σWtrW~TW~+12σWtrWTW-12σDD-~M2+12σDD-M2+ε1+D-MεDFurthermore, the following inequality holds: (65)V˙≤-λV+εwhere(66)λ=min⁡2K1-14I,2K2,1τ,ηDσD,σWε=12σWtrWTW+12σDD-ω2+ε1+D-ωεDSolving (65) implies that 0≤Vt≤maxε/2λ,V0. Therefore, we can conclude that all the signals in the closed-loop system are globally ultimately bounded. Moreover, since si,i∈α,β,γV are bounded, the predefined control performance bounds can be guaranteed. The proof is complete.Remark 9.
Note that the matched uncertainties and the actuator derivations are quickly time-varying. To handle the high dynamic uncertainties, we design the adaptive laws (49) and introduce a novel nonlinear function given in (14). As a result, the robustness of the proposed method with respect to the time-varying uncertainties can be improved. By estimating the upper bounds of the time-varying matched disturbances, this problem can be addressed. Moreover, as is well known, SMDO possess strong robustness and accuracy to estimate a lumped disturbance including unknown external disturbances and parametric uncertainties. As a further development, the second-order sliding mode disturbance observer can produce continuous estimation signals and possess strong robustness. In the dynamic model of the NSV, the mismatched disturbance cannot be neglected. Therefore, to deal with the mismatched uncertainties dmis=Δf1(x1,x2)+DF existing in (24), a second-order sliding mode disturbance observer is introduced for estimation and compensation. At last, it should be noted that the modeling errors are also included in the high dynamic matched uncertainties and the mismatched uncertainties. Based on the previous analysis, it can be concluded that the proposed method has strong robustness with respect to the modeling errors.


## 3.1. Performance Functions

Defineeα=α-αr,eβ=β-βr and eγV=γV-γV,r. Then, for the purpose of guaranteeing the predefined control performance, we introduce the performance functions in the following text. Define the following exponentially decaying functions pit,i∈α,β,γV as the performance functions:(20)pit=pi,0-pi,∞exp⁡-ait+pi,∞,i∈α,β,γVwhere pi,0>pi,∞>0 and ai>0. Define b_i,b-i>0. If the inequality -b_ipi(t)<ei(t)<b-ipi(t) holds for t≥0, then the transient performance is ensured. To make this point more clear, we make the following explanations. b-ipi,0 and -b_ipi,0 restrict the range of the overshoot and the lower bound of the undershoot of eit. The convergence speed of eit is restrained by the decreasing speed of pit. The steady error eit is constrained by max⁡{b_ipi,∞,b-ipi,∞}.Consider the following strictly increasing function:(21)si=Teipi=ln⁡b_i+eipi-ln⁡b-i-eipi,i∈α,β,γVClearly, function (21) satisfies(22)limei/pi→-b_i⁡Teipi=-∞,limei/pi→b-iTeipi=+∞It is not difficult to verify that the inverse function S-1∙ is well-defined and strictly increasing as well. Obviously, if si is bounded, eit/pit remains within a compact subset of -b_i,b-i. As a result, the control issue forces si to be bounded.


## 3.2. Second-Order Sliding Mode Disturbance Observer

Definex1,r=αr,βr,γV,rT and e1=x1-x1,r. Consider the NSV model given in (9); we can get the following dynamic equation: (23)e˙1=Ax2+f1x1,x2+Δf1x1,x2+DF-x1,rx˙2=f2x1,x2+Δf2x1,x2+Bu+BΔδ+DMUsing the strictly increasing function provided in (21), it can be proven that (24)s˙=s-Ax2+f1x1,x2+Δf1x1,x2+DF-x1,r-ψx˙2=f2x1,x2+Δf2x1,x2+Bu+BΔδ+DMwhere (25)s=sα,sβ,sγVT,s-=diag⁡s-α,s-β,s-γVψ=ψα,ψβ,ψγVTs-i=∂T∂ei/pi1pi,ψi=eip˙ipi,i∈α,β,γVConsider the mismatched uncertainties dmis=Δf1(x1,x2)+DF existing in system, a second-order sliding mode disturbance observer is introduced for estimation and compensation. The SOSMDO is designed as (26)d^mis=v,v=-k1ξξ1/2+z-k2ξ,z˙=-k3ξξ-k4ξξ=z0-e1,z˙0=Ax2+f1x1,x2-x1,r+vIn this work, we divide the mismatched uncertainties into two parts. One is the unknown disturbance related to the system states; the other is the time-varying unknown uncertainty. In other words, (27)dmis=d1t,ξ+∫0td2τdτWithout loss of generality, it is supposed that there exist constants g1,g2 such that (28)d1t,ξ≤g1ξ,d2≤g2Theorem 7.
Consider system (23). If one selects k1,⋯,k4 such that(29)k1>2g2,k2>2g1,k3>max⁡k-3Θ,k-3Υ,k4>max⁡k-4Θ,k-4Υ,where (30)k-3Θ=3g2+2g22k12,k-4Θ=3/2k12k2+3g2k22k3k12-2g22-3g2k12+k22+32k2g1k-3Υ=9/16k1g12k2k2-2g1+1/2k12g1-2k12k2+k2g2k2-2g1k-4Υ=1k2-2g19/16k1g12k2+1/2g12/k22k2k3+2k12-g2-2k3+1/2k12g1-9/16k1g12+2k22g1+1/4k2g12k2-2g1 then the estimation error of the mismatched uncertainties can be forced to zero in finite time.Proof.
From (23) and (26), we can get the following closed-loop dynamic equation:(31)ξ˙=-k1ξξ1/2+z-k2ξ-dmisz˙=-k3ξξ-k4ξ Consider the following Lyapunov function:(32)Vξ,z=2k3ξ+k4ξTξ+12zTz+ηTηwhere η=k1ξξ-1/2+k2ξ-z. Taking the derivatives of (32) yields(33)V˙ξ,z=2k3+k122ξTξ˙ξ+2k122+k4ξTξ˙+2zTz˙+32k1k2ξTξ˙ξ1/2-k2ξ˙Tz+ξTz˙-k1-12ξTξ˙zTξξ5/2+z˙TξzTξ˙ξ1/2 Substituting (31) into (33) yields(34)V˙ξ,z=-k1k3+k132ξ2ξ3/2+32k1k2ξTd1ξ1/2-k2k4+k23ξ2-k4k1+52k1k22ξ2ξ1/2+k12ξTzξ+2k22ξTz+3k1k2ξTzξ1/2-k2z2+k12ξTzzTξξ5/2-k1zTzξ1/2+2k3+k122ξTd1ξ+2k4+k22ξTd1-k3k2+2k12k2ξ2ξ-k2d1Tz+k12ξTd1zTξξ5/2-k1zTd1ξ1/2+2zTd2-k2ξTd2-k1d2Tξξ1/2 Furthermore, (35)V˙ξ,z≤-k1k3+k132ξ1/2-k3k2+2k12k2ξ-k2k4+k23ξ2-k4k1+52k1k22ξ3/2+k12ξTzξ+2k22ξTz+3k1k2ξTzξ1/2-k2z2+k12ξTz2ξ5/2+2k3+k122ξTd1ξ+2k4+k22ξTd1+32k1k2ξTd1ξ1/2+k2d1Tz+k12ξTd1zTξξ5/2+k1zTd1ξ1/2+2zTd2+k2ξTd2+k1d2Tξξ1/2By using (28) we can get that (36)V˙ξ,z≤-k1k3+k132ξ1/2-k3k2+2k12k2ξ-k4k1+52k1k22ξ3/2+k12z-k2k4+k23ξ2+2k22ξz+3k1k2ξ1/2z-k2z2+k12z2ξ1/2+2k3+k122g1ξ+2k4+k22g1ξ2+32k1k2ξ3/2g1+k2g1ξz+32k1g1ξ1/2z+2g2z+k2g2ξ+k1g2ξ1/2Denote ζ=colξ1/2,ξ,z; then we can rewrite (36) as(37)V˙≤-1ξ1/2ζTΘζ-ζTΥζwhere(38)Θ=Θ110Θ130Θ22Θ23Θ31Θ32Θ33,Υ=Υ110Υ130Υ22Υ23Υ31Υ32Υ33,Θ11=12k13+k1k3-g2k1,Θ13=-12k12-g2,Θ22=k4k1+52k22k1-32k1k2g1,Θ23=-32k1k2,Θ31=Θ13,Θ32=Θ23,Θ33=12k1,Υ11=k2k3+2k12k2-k2g2-2k3+12k12g1,Υ13=-34k1g1,Υ22=k4k2+k23-k22+2k4g1,Υ23=-k22-12k2g1,Υ31=Υ13,Υ32=Υ23,Υ33=k2Clearly, by using condition (29),we can easily get that Θ>0 and Υ>0. Then, it follows from Rayleigh’s inequality that(39)V˙≤-1ξ1/2ζTΘζ≤-1ξ1/2λminΘζ2By defining ζ-=colξξ-1/2,ξ,z, we can rewrite (39) as(40)V˙≤-1ξ1/2λminΘζ-2Furthermore, (41)V˙≤-1ξ1/2λminΘλmaxΥVSince V1/2>λminΥξ1/2, it can be proven that (42)V˙≤-λminΘλminΥλmaxΥV1/2According to Lemma 3, it can be observed that the equilibrium point ξ,z=0 can be reached in finite time. Then by substituting ξ=z=0 into (31), we know ξ˙=0. As a result, ξ=ξ˙=0 is achieved in finite time. Define the estimation error of the mismatched uncertainty as d~mis=d^mis-dmis. Then from (26), it can be proven that ξ˙=d~mis. Therefore, it can be concluded that d~mis is forced to zero in finite time. The proof is completed.


## 3.3. Hierarchical Fuzzy Approximation-Based Adaptive Fault-Tolerant Tracking Control

Considering the following transformed dynamic equation(43)s˙=s-Ax2+f1x1,x2+Δf1x1,x2+DF-x1,r-ψx˙2=f2x1,x2+Δf2x1,x2+Bu+BΔδ+DMwe design the virtual control signal as follows: (44)x2c=s-A-1-K1s-s-f1x1,x2-s-d^mis+s-x˙1,r+s-ψwhere K1∈R3×3. To avoid the excessive complexity of control design, a first-order filter is established as(45)τx˙2d+x2d=x2c-τATs-Ts,x2d0=x2c0where τ∈R3×3 is the parameter matrix of the filter. Denote e2=x2-x2d,y=x2d-x2c. Then from (43) ~ (45) we can infer the closed-loop dynamic equation as(46)s˙=s-Ae2+y-K1s-s-d~misy˙=-1τy-ATs-Ts-x˙2cAccordingly, along (43), take the derivate of e2 as(47)e˙2=f2x1,x2+Δf2x1,x2+Bu+BΔδ+DM-x˙2dIn view of the unknown nonlinearities existing in (47), a hierarchical fuzzy logic system WTΦx is constructed for compensation. Ideally, Δf2x1,x2=WTΦx+εϕ. W is the weight matrix of the HFLS, N is the number of the rules, and Φx∈RN is the membership function. The final control law is developed as(48)u=B-1-K2e2-f2x1,x2-W^TΦx+x˙2d-ATs-Ts-D-^Me2e2Te2+εD2where K2=diag⁡k21,k22,k23; W^∈RN×3 is the estimation of W. D-^M represent the estimations of D-M, D-M=supt≥0DM+BΔδ+εϕ. The adaptive laws are given by (49)D-^˙M=ηDe2Te2e2Te2+εD2-σDD-^MW^˙=ΓΦxe2T-σWW^σD,ηD,σW>0 are design constants. Γ>0 are the gain matrixes of the adaptive laws. The structure of the proposed control method is provided in Figure 2.Figure 2
The structure of the proposed control method.Theorem 8.
Consider system (43). Suppose the estimation error of the mismatched uncertainty is bounded. Then, by using control laws (44) and (48) and adaptive laws (49), it can be ensured that all signals of the overall closed-loop system are globally uniformly bounded. As a result, the predefined control performance bounds can be guaranteed.Proof.
Consider the following Lyapunov function candidate:(50)V=V1+V2V1=12sTs+12yTyV2=12e2Te2+12ηDD-~MD-~M+12trW~TΓ-1W~Then, along (46), the derivative of V1 can be taken as(51)V˙1=sTs-Ae2+y-sTK1s-sTs-d~mis-yTyτ-yTATs-Ts-yTx˙2cResorting to Young’s inequality, it can be proven that (52)-yTx˙2c≤yTy2τ+τεc2-sTs-d~mis≤14sTs+ε-mis2where ε-mis=supt≥0s-d~mis,εc=supt≥0x˙2c. Therefore, we can rewrite (51) as(53)V˙1=sTs-Ae2-sTK1-14Is-yTy2τ+ε1where (54)ε1=ε-mis2+τεc2 From (47) and (48), it can be obtained that (55)e˙2=-K2e2-D-^Me2e2Te2+εD2-W~TΦx-ATs-Ts+BΔδ+DM+εϕwhere G~=G^-G, D~ω=D-^ω-D-ω. Take the derivative of V2 along (55) as(56)V˙2=-e2TK2e2-e2TATs-Ts-e2TD-^ωe2e2Te2+εD2-e2TW~TΦx+e2TBΔδ+DM+εϕ+1ηDD-~MD-^˙M+trW~TΓ-1W^˙By using Lemma 4, we can get that (57)e2TBΔδ+DM+εϕ≤e2D-M≤D-Me2Te2e2Te2+εD2+D-MεDFurther(58)-e2TD-^Me2e2Te2+εD2+e2TBΔδ+DM+εϕ≤-D-~Me2Te2e2Te2+εD2+D-MεDCombining (56) ~ (58) yields (59)V˙2≤-e2TK2e2-e2TATs-Ts-D-~ωe2Te2e2Te2+εD2-e2TW~TΦx+1ηDD-~MD-^˙M+trW~TΓ-1W^˙+D-MεDBy using the vector trace identity aTb=trbaT we can obtain that (60)V˙2≤-e2TK2e2-e2TATs-Ts-D-~ωe2Te2e2Te2+εD2-e2TW~2TΦ2x+1ηDD-~ωD-^˙ω+trW~TΓ-1W^-Φxe2T+D-MεDSubsisting (49) into (60), we come to(61)V˙2≤-e2TK2e2-e2TATs-Ts-σDD-~MD-^M-σWtrW~TW^+D-MεDThen it follows from (53) and (61) that (62)V˙≤-e1TK1-14Ie1-e2TK2e2-yTy2τ-σWtrW~TW^-σDD-~MD-^M+ε1+D-MεDClearly, (63)-W~TW^≤-12W~TW~+12WTW,-D-~ωD-^ω≤-12D-~ω2+12D-ω2Therefore, it can be checked that (64)V˙≤-e1TK1-14Ie1-e2TK2e2-yTy2τ-12σWtrW~TW~+12σWtrWTW-12σDD-~M2+12σDD-M2+ε1+D-MεDFurthermore, the following inequality holds: (65)V˙≤-λV+εwhere(66)λ=min⁡2K1-14I,2K2,1τ,ηDσD,σWε=12σWtrWTW+12σDD-ω2+ε1+D-ωεDSolving (65) implies that 0≤Vt≤maxε/2λ,V0. Therefore, we can conclude that all the signals in the closed-loop system are globally ultimately bounded. Moreover, since si,i∈α,β,γV are bounded, the predefined control performance bounds can be guaranteed. The proof is complete.Remark 9.
Note that the matched uncertainties and the actuator derivations are quickly time-varying. To handle the high dynamic uncertainties, we design the adaptive laws (49) and introduce a novel nonlinear function given in (14). As a result, the robustness of the proposed method with respect to the time-varying uncertainties can be improved. By estimating the upper bounds of the time-varying matched disturbances, this problem can be addressed. Moreover, as is well known, SMDO possess strong robustness and accuracy to estimate a lumped disturbance including unknown external disturbances and parametric uncertainties. As a further development, the second-order sliding mode disturbance observer can produce continuous estimation signals and possess strong robustness. In the dynamic model of the NSV, the mismatched disturbance cannot be neglected. Therefore, to deal with the mismatched uncertainties dmis=Δf1(x1,x2)+DF existing in (24), a second-order sliding mode disturbance observer is introduced for estimation and compensation. At last, it should be noted that the modeling errors are also included in the high dynamic matched uncertainties and the mismatched uncertainties. Based on the previous analysis, it can be concluded that the proposed method has strong robustness with respect to the modeling errors.


## 4. Simulation Study

In this section, the proposed adaptive fuzzy tracking controller is used for the attitude system of the NSV and simulation results are given to illustrate the effectiveness. The conventional model reference adaptive control method (MRAC) [41] and the terminal sliding mode control method (TSMC) [42] are employed in the simulation to verify the advantages of the proposed method. In the simulation, we consider two cases of the multiple uncertainties and actuator faults.Firstly, we consider different levels of multiple uncertainties before and after the instantt=5s in the simulation. The unknown winds, which can induce aerodynamic disturbances, are taken into consideration. Let U and W represent the speed of the horizontal wind and the vertical wind, respectively. When t≤5s, the unknown winds are set as U=30m/s,W=60m/s and when t≥5s they turn to U=40m/s,W=70m/s. The perturbations of aerodynamic parameters are set as -20% when t≤5s and -30% when 5s<t≤20s. The actuator deviations are set as 0.30.60.4T when t≤5s and 1.10.90.95T when 5s<t≤20s. The parameters of the SOSMDO are set as k1=7,k2=7,k3=10,k4=10. Meanwhile, we choose K1=diag⁡3,3,3,K2=diag⁡8,8,8 and σD=0.05,σW=0.05. The filter parameters are set as τ=0.08. Figures 3–8 provide the simulation results.Figure 3
The trajectories of angle of attack under the three methods of Case 1.Figure 4
The trajectories of sideslip angle under the three methods of Case 1.Figure 5
The trajectories of bank angle under the three methods of Case 1.Figure 6
The trajectories of the control signals under the three methods of Case 1.Figure 7
The trajectories of the adaptive parameters of Case 1.Figure 8
The estimation results of the mismatched uncertainties using SOSMDO.The tracking performance of the attitude angles is given in Figures3–5. It is easy to find that the proposed method can achieve satisfactory control performance in the presence of actuator faults and multiple uncertainties. The proposed method can steer the tracking errors to zero, while the MRAC and TSMC methods cannot. The control signals and adaptive parameters are given in Figures 6 and 7, which illustrate the boundness of the signals of the closed-loop system. The estimation results for the mismatched uncertainties via SOSMDO are shown in Figure 8. The exact estimation for the mismatched disturbances improves the control accuracy.Then we consider the sinusoidal uncertainties and actuator faults. Whent≤5s and 5s<t≤20s, it is assumed that the aerodynamic parameters possess 0.1cos⁡t and 0.2cos⁡t uncertainties, respectively. The parameters of the other uncertainties and actuator deviations are provided in Table 1. We select the control gains, observation gains, and the adaptive gains as same as Case 1. The simulation results are given in Figures 9–12. It is obvious that the satisfactory attitude tracking performance can also be achieved when the NSV suffers from sinusoidal external disturbances. The strong robustness of the proposed method with respect to the different kinds of multiple uncertainties is demonstrated therefore.Table 1
The parameters of the uncertainties and actuator deviations of Case 2.
Variable t ≤ 5 s 5 s < t ≤ 20 s U m / s 20 + 2 cos ⁡ 0.5 t 50 + 5 cos ⁡ 0.5 t W m / s 50 + 3 sin ⁡ 0.2 t 80 + 7 sin ⁡ 0.2 t Δ δ i 0.05 cos ⁡ 0.5 t 0.25 cos ⁡ 0.5 tFigure 9
The trajectories of angle of attack under the three methods of Case 2.Figure 10
The trajectories of sideslip angle under the three methods of Case 2.Figure 11
The trajectories of bank angle under the three methods of Case 2.Figure 12
The trajectories of the control signals under the three methods of Case 2.


## 5. Conclusions

In this paper, an adaptive tracking controller based on hierarchical fuzzy approximator and adaptive bound estimation approach has been developed to achieve high-performance attitude control for the NSV. By introducing a second-order sliding mode disturbance observer, the time-varying mismatched uncertainties can be estimated and compensated. A new nonlinear function is employed to handle the high dynamic matched uncertainties. By using the performance functions and transforming the original system into one incorporating the desired performance criteria, the predefined tracking performance can be guaranteed. Finally, simulation results illustrate that the proposed control scheme can achieve satisfactory performance under the disturbance environment.



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*Source: 4253971-2018-07-10.xml*