# Neural Network-Based Output Feedback Fault Tolerant Tracking Control for Nonlinear Systems with Unknown Control Directions

Authors: Kun Yan; Chaobo Chen; Xiaofeng Xu; Qingxian Wu
Journal: Complexity
 (2022)

---

## Abstract

In this study, an adaptive output feedback fault tolerant control (FTC) scheme is proposed for a class of multi-input and multioutput (MIMO) nonlinear systems with multiple constraints. The neural network (NN) is adopted to handle the unknown nonlinearity by means of its superior approximation capability. Based on it, the state observer is designed to estimate the unmeasured states, and the nonlinear disturbance observer is constructed to tackle the external disturbances. In addition, the Nussbaum function is utilized to cope with the actuator faults, which are coupled with the unknown control directions. Combining with the Lyapunov theory, a NN-based output feedback FTC law is developed for the MIMO nonlinear systems, and the boundedness of all closed-loop system error signals is proved. Simulation results on the unmanned helicopter are performed to demonstrate the effectiveness of the proposed controller.


---

## Body

## 1. Introduction

During these years, the control issue of multi-input and multioutput (MIMO) nonlinear systems has been drawing great attention due to their generalized and natural description for many practical applications, such as ocean vessel system, mobile robot system, and unmanned aerial vehicle system [1–4]. Since existing linear control methods do not match with the nonlinear systems, many effective nonlinear control approaches have been proposed and considerable results have been received. In [5–9] and the references therein, the adaptive backstepping control and sliding mode control schemes were, respectively, developed for a class of nonlinear systems to achieve acceptable control performance. Nevertheless, as the system structure becomes more and more complex, the unknown nonlinearity caused by nonlinear elements always appears and is worthy of further study.Nowadays, in order to handle the unknown nonlinearity in the nonlinear systems, numerous control strategies have been presented, such as the neural network (NN) [10–13], extended state observer [14, 15], and fuzzy logic system [16–18]. In particular, the NNs are often combined with other control techniques to deal with the specific nonlinear systems due to their unique properties. In [10, 11], the adaptive backstepping-based neural control schemes were severally proposed to handle the unknown nonlinear functions for the helicopter and near space vehicle (NSV) systems. In [12], the NNs were combined with the model predictive control to overcome the so-called quadratic programming problem for a class of nonholonomic chained systems. In [13], a robust NN sliding mode control law was developed for a two-axis motion control system with unknown nonlinear terms and external disturbances. However, plenty of the above-mentioned literature are based on the available states, which are strict for some practical control systems. Actually, when the sensors malfunction, the signal usually cannot be accurately measured, and the state feedback approach is unreliable. Therefore, the high-quality controller needs to be further designed when the MIMO systems suffer from unmeasurable states.The state observer is a common method to estimate the unmeasurable states in the academic community [19]. Meanwhile, fruitful and significant results can be found in the recent literature [20–26]. In [20], an adaptive NN output feedback optimized controller was designed for the nonlinear systems with unknown dead zones. In [21, 22], the output feedback control problem for uncertain stochastic nonlinear systems was investigated by employing the state observer approach. In [23], an observer-based finite time controller was developed for a class of second-order nonlinear homogenous systems. Considering the objective existence of performance constraint, the output feedback control schemes were separately presented for the single-input and single-output and MIMO nonlinear systems in [24, 25]. In [26], a stable fuzzy output feedback controller was constructed for a class of nonlinear systems by means of the small-AAIN approach. Nevertheless, it is worth noting that the control directions are usually unknown, and actuator faults occur frequently in many practical application [27, 28]. Therefore, for the purpose of maintaining admissible system performance, more attentions and deeper considerations should be paid to the issues of unknown control directions and actuator faults.At present, Nussbaum gain theory, which was first proposed by Nussbaum in 1983 [29], has become one of the most valid methods to cope with the unknown control directions. Meanwhile, existing achievements show that the combination of Nussbaum gain control technology with nonlinear control method attains favorable control performance in practice [30–33]. In [30], a distributed consensus controller for uncertain nonlinear systems with unknown control directions was designed to complete control task. In [31], the adaptive protocol was presented for nonlinear systems when both control directions and parameters were all unknown. In [32], a predefined performance-based adaptive controller was proposed for MIMO nonlinear systems in presence of unknown control directions and unknown hysteresis nonlinearities. In [33], an adaptive fuzzy tracking controller was designed to assure the stability of the stochastic nonlinear systems. However, the actuator faults are neglected in the above literature, whose occurrence will worsen the system performance and even lead to instability. Over the years, a quantity of attentions have been focused on fault tolerant control (FTC), and some effective control strategies have been implemented to faulty system. In [34], a robust adaptive sliding mode FTC scheme was proposed for coaxial helicopter to deal with actuator faults. In [35], a fuzzy adaptive nonlinear FTC strategy was presented for hypersonic vehicles with actuator stuck and loss of effectiveness faults. In [36], an adaptive decentralized FTC algorithm was developed for NSV attitude dynamics with actuator faults and control surface damage. Nonetheless, besides the negative factors mentioned above, the time-varying disturbances derived from the outside world also should be further considered.The problem of robust control has a long history, and a number of disturbance rejection methods have been developed in past decades [15, 19, 37–40]. Especially, the nonlinear disturbance observer (NDO) has been extensively employed in practice since it does not depend on complete information of the disturbance model. In [15], the NDO-based composite fuzzy control issue was investigated when the uncertain nonlinear systems suffered from unknown dead zone. In [19], a NDO-based output feedback control scheme was developed for uncertain nonlinear systems with unknown hysteresis and external disturbance. By combining the NDO and asymptotic tracking control techniques, a composed control approach was proposed for the spacecraft formation flying system under nonzero disturbances in [38]. However, when the unknown nonlinearity, unknown control directions, unmeasurable states, actuator faults, and external disturbances appear simultaneously, the control performance of the MIMO nonlinear system faces severe challenges, and it is of great significance to design high-quality control algorithms.Motivated by above analysis, the radial basis function neural network (RBFNN), state observer, Nussbaum function, and NDO are combined with the backstepping technique to achieve satisfactory tracking control property. The main contributions can be summarized as follows:(1)
Different from the traditional state feedback control [7], an output feedback controller is designed to tackle the unmeasured states and unknown nonlinearity by means of the RBFNN and state observer;(2)
Compared with some direct adaptive fault estimation method using projection function [11], the presented Nussbaum-based FTC approach can overcome the singularity problem in a simpler way and reduce the complexity of controller design;(3)
The developed output feedback FTC scheme can guarantee satisfactory tracking performance for the MIMO nonlinear systems under multiple negative effects.The rest of this paper is organized as follows. Problem formulation and preparation knowledge are presented in Section2. Section 3 derives the main results. Simulation studies on unmanned helicopter are carried out in Section 4. Section 5 draws the conclusion.


## 2. Problem Formulation and Preparation

Consider a class of MIMO nonlinear systems with actuator faults and external disturbances as follows:(1)X˙i=FiX¯i+Xi+1+Dit,i=1,2,…,n−1,X˙n=FnX¯n+Bϑu+Dnt,y=X1,where Xi∈Rn and X¯i=X1T,X2T,…,XiTT∈Rin with i=1,2,…,n being the system state vectors. u∈Rn and y∈Rn denote the input vector and output vector, respectively. FiX¯i∈Rn defines an unknown smooth nonlinear function. B=diagB1,B2,…,Bn is the unknown nonzero constant control gain matrix. ϑ=diagϑ1,ϑ2,…,ϑn, ϑi refers to the actuator fault factor, which describes the unknown remaining control efficiency of ith actuator. Dit represents external time-varying disturbance. In this paper, it is assumed that only the system output y is available for measurement.Remark 1.
The considered MIMO nonlinear systems can be employed to describe many practical plants, such as ocean vessel system [2], unmanned helicopter system [4], and NSV system [11]. Moreover, it is worth mentioning that the control directions are unknown in many application requirements and usually coupled with actuator faults. Hence, for reflecting the system dynamics more practically, the unknown control directions and actuator faults are considered simultaneously in this work.
The control objective of this study is to design a robust adaptive NN output feedback FTC scheme, such that all error signals of the closed-loop system are convergent, and the outputy can follow the desired trajectory yd. To this end, the following definition, lemmas, and assumptions are introduced.Definition 1.
(see [33]). If a continuous function H⋅ fulfills the performances as follows:(2)lims⟶+∞sup1s∫0sHτdτ=+∞,lims⟶−∞inf1s∫0sHτdτ=−∞.
ThenH⋅ is called a Nussbaum function. At present, the functions τ2cosτ, eτ2cosτ, τ2sinτ, and eτ2sinπ/2τ have been proven to be the Nussbaum-type functions in [41, 42]. In this paper, Hτ=τ2cosτ is used.Lemma 1.
(see [33]). Let Vt≥0 and τit be smooth functions defined on 0,tf, and let Hτi be smooth Nussbaum-type function. If the following inequality holds:(3)V˙t≤−b0Vt+b1+∑i=1nψiHτi+1τ˙i,where b0, b1 and ψi are suitable positive constants, then, Vt, τit and ∑i=1nψiHτi+1τ˙i must be bounded on 0,tf.Lemma 2.
(see [10]). Owing to the powerful nonlinear approximation capability, RBFNN is frequently employed to approximate any unknown smooth nonlinear function Eγ, which can be expressed as(4)Eγ=Φ^TMγ+σ,where γ∈Rn is the input vector, Φ^∈Rj×n is the weight matrix, σ is the approximation error, and Mγ=Mγ,M2γ,…,Mjγ∈Rj is the basis function vector with Mjγ being(5)Mjγ=exp−γ−νiTγ−νici2,i=1,2,…,j,where νi and ci denote the center and width of the basis function, respectively.Then, the RBFNN (4) can approach any unknown smooth function Eγ in the form of(6)Eγ=Φ∗TMγ+σ∗,where Φ∗ is the optimal weight matrix, σ∗ is the approximation error satisfying σ∗≤σ¯, and σ¯ is a positive constant.Assumption 1.
(see [7]). For the bounded desired trajectory yd and its derivatives y˙d and y¨d, there exists an unknown positive constant l¯0 making Π0:=yd,y˙d,y¨d:yd2+y˙d2+y¨d2≤l¯0 holds.Assumption 2.
(see [11]). The unknown external disturbance Di,i=1,2…,n is supposed to satisfy Di≤D¯i and D˙i≤ζi, where D¯i and ζi are unknown positive constants.Assumption 3.
(see [11, 33]). The control gain Bi is assumed to have the unknown but same sign with each other, and it satisfies Bl≤Bi≤Bu, where Bl>0 is the known constant. Moreover, the fault factor ϑi is assumed to be unknown constant and satisfies 0<ϑm≤ϑi≤1, where ϑm is the known lower bound.Remark 2.
For a practical system, the tracking mission should be realizable, and there should exist a feasible controller to achieve it, which means that assumption1 is reasonable. In addition, assumption 2 is provided to illustrate that if the external disturbance is unbounded, it may result in that the system cannot provide enough energy to accomplish the specified object. Similarly, if the actuator loses too much effectiveness, the whole system may lose the FTC capacity. Therefore, it is reasonable for assumptions 1–3, and they have been extensively used in existing literature such as [4, 5, 7, 16–18, 40–42].


## 3. Control Design and Stability Analysis

In this section, a backstepping-based robust adaptive output feedback FTC control scheme will be developed for the MIMO nonlinear systems to deal with the unknown nonlinearity, unavailable states, unknown control directions, actuator faults, and external disturbances. The block diagram of the design thread is given in Figure1.Figure 1
Block diagram of the proposed control scheme.


### 3.1. Model Transformation and State Observer Design

Since the control gainBi and fault factor ϑi of the ith actuator are all unknown and coupled with the corresponding control input, it is difficult to design the controller directly. According to the properties of Bi and ϑi, it can be obtained that 0<Blϑm≤Biϑi≤Bu. By defining ℏ=Bϑ, it is known that ℏ is an invertible matrix. In order to promote the control design, we introduce the new state variable xi=ℏ−1Xi, the new smooth function fix¯i=ℏ−1FiX¯i, and the new disturbance dit=ℏ−1Dit. Then, the MIMO nonlinear system (1) can be transformed into the following system:(7)x˙i=fix¯i+xi+1+dit,i=1,2,…,n−1,x˙n=fnx¯n+u+dnt,y1=x1,where x¯i=x1T,x2T,…,xiTT∈Rin and y1∈Rn are the new system state vector and output vector.Remark 3.
Considering assumption2 and assumption 3, we obtain that the new disturbance dit and its first derivative are also bounded. That is, there exist unknown positive constants d¯i and ϱi such that di≤d¯i and d˙i≤ϱi. Moreover, we should note that the state variable xi and the output vector y1 of the new MIMO nonlinear system (7) are all unavailable.
Here, the following RBFNN is adopted to approximate the unknown nonlinear termLifix¯i:(8)Lifix¯i=Φi∗TMix¯i+σi∗x¯i,i=1,2,…,n,where Li=LiT>0 is the designed constant matrix, Mix¯i is the basis function, and Φi∗ is the weight matrix.
Due to the unavailability of the system statesxi, the above function approximations (8) are invalid. Hence, we use the following approximation to design the state observer:(9)Lif^ix¯^i=Φ^iTMix¯^i,i=1,2,…,n,where x¯^i=x^1T,x^2T,…,x^iTT is the estimation of x¯i, and Φ^i is the estimation of Φi∗.
It is obvious that the approximationLif^ix¯^i depends on the available Φ^i and x¯^i. In order to cope with the unknown system states in the MIMO nonlinear systems (7), the following state observer is designed [19]:(10)x^˙i=x^i+1+f^ix¯^i−kix^1+χiy+d^i,i=1,…,n−1,x^˙n=u+f^nx¯^n−knx^1+χny+d^n,where ki=diagki1,ki2,…,kin with kij>0, χi=diagχi1,χi2,…,χin with χij>0j=1,2,…n, d^i is the estimation of di. The corresponding updating laws with respect to d^i and Φ^i will be given gradually along with the controller design process in the next subsection.
Definezi=xi−x^i, λi=Li−1σi∗x¯i, and Φ˜i=Φi∗−Φ^i. Considering (7–10) and differentiating zi yield(11)z˙i=xi+1+Li−1Φi∗TMix¯i+σi∗x¯i+di−x^i+1−Li−1Φ^iTMix¯^i+kix^1−χiy−d^i,=−kiz1+zi+1+kix1+λi−χiy+Li−1Φ˜iTMix¯^i+Li−1Φi∗TMix¯i−Mix¯^i−εizi+Δ˜i,(12)z˙n=−knz1+knx1+λn−χny+Ln−1Φ˜nTMnx¯^n+Ln−1Φn∗TMnx¯n−Mnx¯^n−εnzn+Δ˜n,where εi=εiT>0 is the designed diagonal matrix, and Δ˜i is the estimation error of the auxiliary variable Δi, which will be given in the following.
Then, the observation error dynamics (11) and (12) can be rewritten as(13)Z˙=NZ+Kx1+L−1Φ∗TΥx¯,x¯^+λ+Δ˜+L−1Φ˜TMx¯^−hy.where N=−k1−ε1I1……−k2−ε2I2⋮⋮⋮⋱In−1−kn……−εn, Z=z1T,z2T,…,znTT, λ=λ1T,λ2T,…,λnTT, K=k1T,k2T,…,knTT, L−1=diagL1−1,L2−1,…,Ln−1, Φ∗=diagΦ1∗,Φ2∗,…,Φn∗, Φ˜=diagΦ˜1,Φ˜2,…,Φ˜n, Δ˜=Δ˜1T,Δ˜2T,…,Δ˜nTT, h=χ1T,χ2T,…,χnTT, Mx¯^=M1Tx¯^1,M2Tx¯^2,…,MnTx¯^nT, Ii∈Rn×n is the unit matrix, Υx¯,x¯^=Mx¯−Mx¯^, Mx¯=M1Tx¯1,M2Tx¯2,…,MnTx¯nT.
Here, proper parameterski should be selected to ensure that N is Hurwitz. In other words, for a given matrix P=PT>0, there exists positive definite matrix Λ=ΛT>0 such that(14)NTP+PN≤−Λ.
Due to the unknown disturbancedi, the following auxiliary variable Δi is introduced:(15)Δi=di+εixi,i=1,2,…,n.
In light of (15), the NDOs are designed as(16)d^i=Δ^i−εix^i,i=1,2,…,n−1,Δ^˙i=εif^ix¯^i+x^i+1−εix^i+Δ^i,.(17)d^n=Δ^n−εnx^n,Δ^˙n=εnf^nx¯^n+u−εnx^n+Δ^n..
DefineΔ˜i=Δi−Δ^i and d˜i=di−d^i. Then, we have(18)Δ˜i=di+εixi−d^i−εix^i=d˜i+εizi.
On the one hand, by invoking (15) and (16), we obtain(19)Δ˜˙i=d˙i+εix˙i−Δ^˙i=d˙i+εiLi−1Φi∗Tϒix¯i,x¯^i+Li−1Φ˜iTMix¯^i+λi+zi+1−Δ˜i−εizi,i=1,2,…,n−1.
On the other hand, by invoking (15) and (17), we obtain(20)Δ˜˙n=d˙n+εnx˙n−Δ^˙n=d˙n+εnLn−1Φn∗Tϒnx¯n,x¯^n+Ln−1Φ˜nTMnx¯^n+λn−Δ˜n−εnzn.
Consider the following Lyapunov function candidate:(21)V0=ZTPZ+12∑i=1nΔ˜iTΔ˜i.
Invoking (13), (14), (19) and (20), one obtains(22)V˙0≤−ZTΛZ+2ZTPKx1+2ZTPλ+2ZTPΔ˜+2ZTPL−1Φ∗Tϒx¯,x¯^+2ZTPL−1Φ˜TMx¯^−2ZTPhy+∑i=1nΔ˜iTd˙i+∑i=1nΔ˜iTεiLi−1Φi∗Tϒix¯i,x¯^i+∑i=1nΔ˜iTεiLi−1Φ˜iTMix¯^i+∑i=1nΔ˜iTεiλi−∑i=1nΔ˜iTεiΔ˜i−∑i=1nΔ˜iTεi2zi+∑i=1n−1Δ˜iTεizi+1.
Based on Young’s inequality, the following inequalities can be obtained:(23)2ZTPKx1≤ℏmZTPKKTPTZ+e12+ς02,2ZTPλ≤ZTPPTZ+∑i=1nλ¯i22ZTPL−1Φ∗TΥx¯,x¯^≤ZTPPTZ+∑i=1nβi2Li−12Φi∗22ZTPL−1Φ˜TMx¯^≤r1ZTPPTZ+1r1∑i=1nδi2Li−12Φ˜i22ZTPΔ˜≤ZTPPTZ+∑i=1nΔ˜iTΔ˜i−2ZTPhy≤ZTPhhTPTZ+e12+ς02∑i=1nΔ˜iTd˙i≤12∑i=1nΔ˜iTΔ˜i+12∑i=1n〉i2∑i=1nΔ˜iTεiLi−1Φi∗TΥix¯i,x¯^i≤12∑i=1nΔ˜iTΔ˜i+12∑i=1nε¯2Li−12βi2Φi∗2∑i=1nΔ˜iTεiLi−1Φ˜iTMix¯^i≤r22∑i=1nΔ˜iTΔ˜i+12r2∑i=1nε¯2Li−12δi2Φ˜i2,∑i=1nΔ˜iTεiλi≤12∑i=1nΔ˜iTΔ˜i+12∑i=1nε¯2λ¯i2,−∑i=1nΔ˜iTεi2zi≤12∑i=1nΔ˜iTΔ˜i+ε¯42ZTZ,∑i=1n−1Δ˜iTεizi+1≤12∑i=1nΔ˜iTΔ˜i+ε¯22ZTZ,where r1 and r2 are the designed positive constants, yd≤ς0, ℏm=1/Bl2ϑm2, ε¯=maxεi, λi≤λ¯i, Υix¯i,x¯^i≤βi,Mix¯^i≤δi, e1 is the tracking error, which will be given subsequently.
Substituting the above inequalities into (22) produces(24)V˙0≤−ZTΛ−Λ1−ε¯2+ε¯42I1Z+2e12−∑i=1nΔ˜iTεi−7+r22I1Δ˜i+R0+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2,where Λ1=PℏmKKT+hhT+r1+3I1PT, R0=∑i=1n1+1/2ε¯2βi2Li−12Φi∗2+2ς02+∑i=1n1+1/2ε¯2λ¯i2+1/2∑i=1nϱi2.


### 3.2. Robust Adaptive Neural Output Feedback Control Design

Step 1.
Define the following tracking errors:(25)e1=y−yd,ei=x^i−xid,i=2,…,n,where xid is the designed virtual control law.
Considering (7) and taking the time derivative of e1 yield(26)e˙1=X˙1−y˙d=ℏx˙1−y˙d=F1y+ℏx2d+e2+z2+d1−y˙d.
Because of the unknown smooth functionF1y, the following RBFNN is used to approximate it:(27)F1y=QF−1ΦF∗TMFy+σF∗y,where QF=QFT>0 is the designed matrix, MFy is Gaussian function, and ΦF∗ is the weight matrix.
Substituting (27) into (26) follows(28)e˙1=QF−1ΦF∗TMFy+QF−1σF∗y+ℏx2d+ℏe2+ℏz2+ℏd1−y˙d
Consider the Lyapunov function candidate as(29)V1=V0+12e1Te1+12trΦ˜1TΘ1−1Φ˜1+12trΦ˜FTΩF−1Φ˜F,where Θ1=Θ1T>0 and ΩF=ΩFT>0 are the designed matrixes, Φ˜F=ΦF∗−Φ^F.
Invoking (24) and (28), we obtain(30)V˙1≤−ZTΛ−Λ1−ε¯2+ε¯42I1Z+2e12+R0+e1TQF−1ΦF∗TMFy−∑i=1nΔ˜iTεi−7+r22I1Δ˜i+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2+e1TQF−1σF∗y+e1Tℏx2d+e1Tℏe2+e1Tℏz2+e1Tℏd1−e1Ty˙d−trΦ˜1TΘ1−1Φ^˙1−trΦ˜FTΩF−1Φ^˙F≤−ZTΛ−Λ1−ε¯2+ε¯42I1−CZ+e1Tℏx2d+e1TQF−1ΦF∗TMFy−∑i=1nΔ˜iTεi−7+r22I1Δ˜i+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2+5+3Bu22e1Te1+12σ¯F2QF−12+12e2Te2+12d¯12−trΦ˜1TΘ1−1Φ^˙1−trΦ˜FTΩF−1Φ^˙F−e1Ty˙d+R0,where C=diag0n×n,1/2I1,…,0n×n, σF∗≤σ¯F.
Design the virtual control functionx2d and parameter adaptive laws as(31)x2d=Hτμ,(32)τ˙i=e1iμi,i=1,2,…,n,(33)μ=q1e1−y˙d+QF−1Φ^FTMFy−d^1,(34)Φ^˙1=Θ1M1x¯^1e1TL1−1−ϖ1Φ^1,(35)Φ^˙F=ΩFMFye1TQF−1−ϖFΦ^F,where τ=τ1,τ2,…,τnT, μ=μ1,μ2,…,μnT, e1i is the ith element of e1, Hτ=diagHτ1,Hτ2,…,Hτn, Hτi is Nussbaum-type function, which is chosen as Hτi=τi2cosτi, q1=diagq11,q12,…,q1n, q1i, ϖ1 and ϖF are designed positive constants.
Considering (31)–(35), the following facts can be obtained:(36)e1Tℏx2d=∑i=1nℏiHτiτ˙i=∑i=1nℏiHτi+1τ˙i−e1Tμ,(37)e1TQF−1Φ˜FTMFy−trΦ˜FTΩF−1Φ^˙F=trΦ˜FTϖFΦ^F≤−12ϖFΦ˜F2+12ϖFΦF∗2,(38)−trΦ˜1TΘ1−1Φ^˙1=−trΦ˜1TM1x¯^1e1TL1−1−Φ˜1Tϖ1Φ^1≤e1Te1−12ϖ1−δ124L1−1Φ˜12+12ϖ1Φ1∗2,where ℏ=diagℏ1,ℏ2,…,ℏn with ℏi being the diagonal element.
Substituting (36)–(38) into (30) gives(39)V˙1≤−ZTAZ−e1Tϵ1e1−12ϖ1−δ124L1−12Φ˜12−12ϖFΦ˜F2−∑i=2nΔ˜iTεi−7+r22I1Δ˜i+12e2Te2−Δ˜1Tε1−8+r22I1Δ˜1+∑i=1nℏiHτi+1τ˙i+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2+R1,where A=Λ−Λ1−ε¯2+ε¯4/2I1−C−C1, C1=diag1/2I1,0n×n,…,0n×n, ϵ1=q1−9+3Bu2+ε¯2/2I1, R1=R0+1/2ϖFΦF∗2+1/2σ¯F2QF−12+1/2ϖ1Φ1∗2+d¯12.Step 2.
Considering (10) and differentiating e2 give(40)e˙2=x^3+f^2x¯^2−k2x^1+χ2y+d^2−x˙2d=x3d+e3+L2−1Φ^2TM2x¯^2−k2x^1+χ2y+d^2−x˙2d.
In particular, the dynamic surface control technique is employed to avoid the repeated computation ofx˙2d and achieve its available derivative. Let x2d pass the first-order filter η2 [7].(41)h2η˙2+η2=x2d,η20=x2d0,where h2=diagh21,h22,…,h2n>0 is the time constant matrix of the filter.
By definingω2=η2−x2d, we have(42)ω˙2=η˙2−x˙2d=−h2−1ω2+−∂x2d∂ττ˙−∂x2d∂e1e˙1−∂x2d∂ydy˙d−∂x2d∂Φ^FΦ^˙F−∂x2d∂yy˙−∂x2d∂d^1d^˙1,=−h2−1ω2+l2e1,e2,z2,y,y˙d,y¨d,Φ^F,d^1,where l2• is smooth function vector in regard to Π2•. Since the set Π2• is compact, the smooth function l2• has a maximum l¯2 on set Π2• for the given initial conditions.
Considering (41), the virtual control law x3d is designed as(43)x3d=−q2e2+η˙2+k2x^1−χ2y−d^2−L2−1Φ^2TM2x¯^2,where q2=diagq21,q22,…,q2n, q2i is the designed positive constant.
The parameter adaptive law is proposed as(44)Φ^˙2=Θ2M2x¯^2e2TL2−1−ϖ2Φ^2,where Θ2=Θ2T>0 is the designed matrix, and ϖ2>0 is the designed constant.
Invoking (43), we obtain(45)e˙2=−q2e2+e3−h2−1ω2+l2•.
Choose the following Lyapunov function candidate as(46)V2=V1+12e2Te2+12trΦ˜2TΘ2−1Φ˜2+12ω2Tω2.
The time derivative ofV2 is(47)V˙2=V˙1+e2Te˙2−trΦ˜2TΘ2−1Φ^˙2+ω2Tω˙2.
Considering (44), we obtain(48)−trΦ˜2TΘ2−1Φ^˙2=−trΦ˜2TM2x¯^2e2TL2−1−Φ˜2Tϖ2Φ^2≤e2Te2−12ϖ2−δ224L2−12Φ˜22+12ϖ2Φ2∗2.
Substituting (39), (45) and (48) into (47), we have(49)V˙2≤−ZTAZ−∑i=12eiTϵiei−∑i=1212ϖi−δi24Li−12Φ˜i2−12ϖFΦ˜F2−∑i=2nΔ˜iTεi−7+r22I1Δ˜i+e2Te3−Δ˜1Tε1−8+r22I1Δ˜1+∑i=1nℏiHτi+1τ˙i+R2+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2−ω2Th2−1−Iω2,where R2=R1+1/2ϖ2Φ2∗2+l¯22, ϵ2=q2−2I−1/2h2−12I.Step 3.
3≤i≤n−1: Considering (10) and differentiating ei give(50)e˙i=xi+1d+ei+1+Li−1Φ^iTMix¯^i−kix^1+χiy+d^i−x˙id.
Similar to Step2, let xid pass the following first-order filter ηi [7]:(51)hiη˙i+ηi=xid,ηi0=xid0,where hi=diaghi1,hi2,…,hin>0 is the time constant of the filter.
Defineωi=ηi−xid. Differentiating ωi yields(52)ω˙i=−hi−1ωi+−∂xid∂ei−1e˙i−1−∂xid∂ei−2e˙i−2−∂xid∂x^i−1x^˙i−1−∂xid∂Φ^i−1Φ^˙i−1−∂xid∂yy˙−∂xid∂d^i−1d^˙i−1=−hi−1ωi+liei,ei−1,y,x¯^i−1,Φ^i−1,d^i−1,where li• is a smooth function vector in regard to Πi•. Since the set Πi• is compact, the smooth function li• has a maximum l¯i on set Πi• for the given initial conditions.
Considering (51), the virtual control law is developed as(53)xi+1d=−qiei−ei−1+η˙i+kix^1−χiy−d^i−Li−1Φ^iTMix¯^i,where qi=diagqi1,qi2,…,qin, qij,j=1,2,…,n is the designed positive constant.
The parameter adaptive law is designed as(54)Φ^˙i=ΘiMix¯^ieiTLi−1−ϖiΦ^i,where Θi=ΘiT>0 is the designed matrix, and ϖi>0 is the designed constant.
Invoking (53), we obtain(55)e˙i=−qiei−ei−1+ei+1−hi−1ωi+li•.
Select the Lyapunov function candidate as(56)Vi=Vi−1+12eiTei+12trΦ˜iTΘi−1Φ˜i+12ωiTωi,
By invoking (54), it outputs(57)−trΦ˜iTΘi−1Φ^˙i=−trΦ˜iTMix¯^ieiTLi−1−Φ˜iTϖiΦ^i≤eiTei−12ϖi−δi24Li−12Φ˜i2+12ϖiΦi∗2.
Differentiating (56), we obtain(58)V˙i≤−ZTAZ−∑i=1n−1eiTϵiei−∑i=1n−112ϖi−δi24Li−12Φ˜i2−12ϖFΦ˜F2−∑i=2nΔ˜iTεi−7+r22I1Δ˜i+eiTei+1−Δ˜1Tε1−8+r22I1Δ˜1+∑i=1nℏiHτi+1τ˙i+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2+Ri−∑i=2n−1ωiThi−1−I1ωi,where Ri=Ri−1+1/2ϖiΦi∗2+l¯i2, ϵi=qi−3/2I1−1/2hi−12I1.Step 4.
Taking the derivative ofen yields(59)e˙n=u+Ln−1Φ^nTMnx¯^n−knx^1+χny+d^n−x˙nd.
Similarly, letxnd pass the following filter ηn [7]:(60)hnη˙n+ηn=xnd,ηn0=xnd0.where hn=diaghn1,hn2,…,hnn>0 is the time constant of the filter.
Defineωn=ηn−xnd. Taking the derivative of ωn yields(61)ω˙n=−hn−1ωn−∂xnd∂en−1e˙n−1−∂xnd∂en−2e˙n−2−∂xnd∂x^n−1x^˙n−1−∂xnd∂Φ^n−1Φ^˙n−1−∂xnd∂yy˙−∂xnd∂d^n−1d^˙n−1=−hn−1ωn+lnen,en−1,y,x¯^n−1,Φ^n−1,d^n−1,where ln• is smooth function vector in regard to Πn•. Since the set Πn• is compact, the smooth function ln• has a maximum l¯n on set Πn• for the given initial conditions.
Design the actual control input and parameter adaptive law as(62)u=−qnen−en−1+η˙n+knx^1−χny−d^n−Ln−1Φ^nTMnx¯^n,(63)Φ^˙n=ΘnMnx¯^nenTLn−1−ϖnΦ^n,where qn=diagqn1,qn2,…,qnn with qnjj=1,2,…,n being positive constant, Θn=ΘnT>0 is the designed matrix, and ϖn>0 is the designed constant.
Define the Lyapunov function candidate as(64)Vn=Vn−1+12enTen+12trΦ˜nTΘn−1Φ˜n+12ωnTωn.
DifferentiatingVn outputs(65)V˙n=V˙n−1+enTe˙n−trΦ˜nTΘn−1Φ^˙n+ωnTω˙n.
Considering (60)–(63), we obtain(66)e˙n=−qnen−en−1−hn−1ωn+ln•,(67)−trΦ˜nTΘn−1Φ^˙n≤enTen+12ϖnΦn∗2−12ϖn−δn24Ln−12Φ˜n2,where ρn is the designed constant.
Substituting (66) and (67) into (65), we have(68)V˙n≤−ZTAZ−∑i=1neiTϵiei−∑i=1n12ϖi−δi24Li−12Φ˜i2−12ϖFΦ˜F2−∑i=2nΔ˜iTεi−7+r22I1Δ˜i+Rn−Δ˜1Tε1−8+r22I1Δ˜1+∑i=1nℏiHτi+1τ˙i+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2−∑i=2nωiThi−1−I1ωi≤−ZTAZ−∑i=1nϵieiTei−Δ˜1Tε1−8+r22I1Δ˜1−12ϖFΦ˜F2−∑i=2nΔ˜iTεi−7+r22I1Δ˜i+Rn−∑i=1n12ϖi−14+1r1+ε¯22r2δi2Li−12Φ˜i2−∑i=2nωiThi−1−I1ωi+∑i=1nℏiHτi+1τ˙i,where Rn=Rn−1+1/2ϖnΦn∗2+l¯n2, ϵn=qn−3/2I1−1/2hn−12I1.


### 3.3. Stability Analysis

Hereto, the following theorem is proposed to reveal the main results of this work.Theorem 1.
Consider a class of MIMO nonlinear systems (1) satisfying assumption 1–3. The state observer is designed as (10), and the NDOs are developed as (16-17), and the virtual control laws are proposed as (31), (43), and (53). Under the presented output feedback FTC scheme (62), all tracking error signals of the closed-loop system are uniformly ultimately bounded.Proof.
From (68), it is observed that(69)V˙n≤−sVn+Γ+∑i=1nℏiHτi+1τ˙i,where s and Γ are given by(70)s=minλminAλmaxP,2λminϵi,λmin2ε1−8+r2I1,ϖFλmaxΩF−1,λmin2ε1−7+r2I1,2λminhi−1−I1,ϖi−1/2+2/r1+ε¯2/r2δi2Li−12λmaxΘi−1,Γ=∑i=1n1+12ε¯2βi2Li−12Φi∗2+∑i=1n1+12ε¯2λ¯i2+12ϖFΦF∗2+12σ¯F2QF−12+d¯12+2ς02+∑i=1n12ϖiΦi∗2+∑i=2nl¯i2+12∑i=1nϱi2.
According to Lemma1, it can be obtained that Vn and ∑i=1nℏiHτi+1τ˙i are bounded over 0,tf. By defining Γ1=maxt∈0,tf∑i=1nℏiHτi+1τ˙i, we obtain(71)V˙n≤−sVn+Γ¯,where Γ¯=Γ+Γ1.
Integrating (71) over 0,t yields(72)Vn≤Γ¯s+Vn0−Γ¯se−st.
From (72) and the definition of Vn, we obtain that Vn is exponentially convergent, i.e., limt⟶∞Vn=Γ¯/s. Hence, all the error signals of the closed-loop remain bounded. This concludes the proof.Remark 4.
In this paper, the actuator faults, which are coupled with the unknown control gains, are considered, and the Nussbaum function is utilized to eliminate their adverse effect. Compared with some direct adaptive method [11], the developed control algorithm can effectively avoid the problem of singularity and simplify the proof process. Compared with some existing observer-based output feedback control methods [18, 20, 43], the proposed robust FTC scheme can guarantee the closed-loop stability for the MIMO nonlinear system with multiple negative effects simultaneously, including unmeasured states, unknown nonlinearity, unknown control directions, actuator faults, and external disturbances.Remark 5.
In the control design, different design parameters will result in different system performance. Choosing largerϵi will increase the control gains and is beneficial for tracking results within bounds. However, excessive control gains require more control energy and may cause input saturation. Consequently, in order to achieve applicable control performance, the selection of corresponding design parameters should satisfy the following conditions and reach a compromise:(73)A>0,ϵi>0,2ε1−8+r2I1>0,hi−1−I1>0,ϖF>0,ϖi−12+2r1+ε¯2r2δi2Li−12>0.


## 3.1. Model Transformation and State Observer Design

Since the control gainBi and fault factor ϑi of the ith actuator are all unknown and coupled with the corresponding control input, it is difficult to design the controller directly. According to the properties of Bi and ϑi, it can be obtained that 0<Blϑm≤Biϑi≤Bu. By defining ℏ=Bϑ, it is known that ℏ is an invertible matrix. In order to promote the control design, we introduce the new state variable xi=ℏ−1Xi, the new smooth function fix¯i=ℏ−1FiX¯i, and the new disturbance dit=ℏ−1Dit. Then, the MIMO nonlinear system (1) can be transformed into the following system:(7)x˙i=fix¯i+xi+1+dit,i=1,2,…,n−1,x˙n=fnx¯n+u+dnt,y1=x1,where x¯i=x1T,x2T,…,xiTT∈Rin and y1∈Rn are the new system state vector and output vector.Remark 3.
Considering assumption2 and assumption 3, we obtain that the new disturbance dit and its first derivative are also bounded. That is, there exist unknown positive constants d¯i and ϱi such that di≤d¯i and d˙i≤ϱi. Moreover, we should note that the state variable xi and the output vector y1 of the new MIMO nonlinear system (7) are all unavailable.
Here, the following RBFNN is adopted to approximate the unknown nonlinear termLifix¯i:(8)Lifix¯i=Φi∗TMix¯i+σi∗x¯i,i=1,2,…,n,where Li=LiT>0 is the designed constant matrix, Mix¯i is the basis function, and Φi∗ is the weight matrix.
Due to the unavailability of the system statesxi, the above function approximations (8) are invalid. Hence, we use the following approximation to design the state observer:(9)Lif^ix¯^i=Φ^iTMix¯^i,i=1,2,…,n,where x¯^i=x^1T,x^2T,…,x^iTT is the estimation of x¯i, and Φ^i is the estimation of Φi∗.
It is obvious that the approximationLif^ix¯^i depends on the available Φ^i and x¯^i. In order to cope with the unknown system states in the MIMO nonlinear systems (7), the following state observer is designed [19]:(10)x^˙i=x^i+1+f^ix¯^i−kix^1+χiy+d^i,i=1,…,n−1,x^˙n=u+f^nx¯^n−knx^1+χny+d^n,where ki=diagki1,ki2,…,kin with kij>0, χi=diagχi1,χi2,…,χin with χij>0j=1,2,…n, d^i is the estimation of di. The corresponding updating laws with respect to d^i and Φ^i will be given gradually along with the controller design process in the next subsection.
Definezi=xi−x^i, λi=Li−1σi∗x¯i, and Φ˜i=Φi∗−Φ^i. Considering (7–10) and differentiating zi yield(11)z˙i=xi+1+Li−1Φi∗TMix¯i+σi∗x¯i+di−x^i+1−Li−1Φ^iTMix¯^i+kix^1−χiy−d^i,=−kiz1+zi+1+kix1+λi−χiy+Li−1Φ˜iTMix¯^i+Li−1Φi∗TMix¯i−Mix¯^i−εizi+Δ˜i,(12)z˙n=−knz1+knx1+λn−χny+Ln−1Φ˜nTMnx¯^n+Ln−1Φn∗TMnx¯n−Mnx¯^n−εnzn+Δ˜n,where εi=εiT>0 is the designed diagonal matrix, and Δ˜i is the estimation error of the auxiliary variable Δi, which will be given in the following.
Then, the observation error dynamics (11) and (12) can be rewritten as(13)Z˙=NZ+Kx1+L−1Φ∗TΥx¯,x¯^+λ+Δ˜+L−1Φ˜TMx¯^−hy.where N=−k1−ε1I1……−k2−ε2I2⋮⋮⋮⋱In−1−kn……−εn, Z=z1T,z2T,…,znTT, λ=λ1T,λ2T,…,λnTT, K=k1T,k2T,…,knTT, L−1=diagL1−1,L2−1,…,Ln−1, Φ∗=diagΦ1∗,Φ2∗,…,Φn∗, Φ˜=diagΦ˜1,Φ˜2,…,Φ˜n, Δ˜=Δ˜1T,Δ˜2T,…,Δ˜nTT, h=χ1T,χ2T,…,χnTT, Mx¯^=M1Tx¯^1,M2Tx¯^2,…,MnTx¯^nT, Ii∈Rn×n is the unit matrix, Υx¯,x¯^=Mx¯−Mx¯^, Mx¯=M1Tx¯1,M2Tx¯2,…,MnTx¯nT.
Here, proper parameterski should be selected to ensure that N is Hurwitz. In other words, for a given matrix P=PT>0, there exists positive definite matrix Λ=ΛT>0 such that(14)NTP+PN≤−Λ.
Due to the unknown disturbancedi, the following auxiliary variable Δi is introduced:(15)Δi=di+εixi,i=1,2,…,n.
In light of (15), the NDOs are designed as(16)d^i=Δ^i−εix^i,i=1,2,…,n−1,Δ^˙i=εif^ix¯^i+x^i+1−εix^i+Δ^i,.(17)d^n=Δ^n−εnx^n,Δ^˙n=εnf^nx¯^n+u−εnx^n+Δ^n..
DefineΔ˜i=Δi−Δ^i and d˜i=di−d^i. Then, we have(18)Δ˜i=di+εixi−d^i−εix^i=d˜i+εizi.
On the one hand, by invoking (15) and (16), we obtain(19)Δ˜˙i=d˙i+εix˙i−Δ^˙i=d˙i+εiLi−1Φi∗Tϒix¯i,x¯^i+Li−1Φ˜iTMix¯^i+λi+zi+1−Δ˜i−εizi,i=1,2,…,n−1.
On the other hand, by invoking (15) and (17), we obtain(20)Δ˜˙n=d˙n+εnx˙n−Δ^˙n=d˙n+εnLn−1Φn∗Tϒnx¯n,x¯^n+Ln−1Φ˜nTMnx¯^n+λn−Δ˜n−εnzn.
Consider the following Lyapunov function candidate:(21)V0=ZTPZ+12∑i=1nΔ˜iTΔ˜i.
Invoking (13), (14), (19) and (20), one obtains(22)V˙0≤−ZTΛZ+2ZTPKx1+2ZTPλ+2ZTPΔ˜+2ZTPL−1Φ∗Tϒx¯,x¯^+2ZTPL−1Φ˜TMx¯^−2ZTPhy+∑i=1nΔ˜iTd˙i+∑i=1nΔ˜iTεiLi−1Φi∗Tϒix¯i,x¯^i+∑i=1nΔ˜iTεiLi−1Φ˜iTMix¯^i+∑i=1nΔ˜iTεiλi−∑i=1nΔ˜iTεiΔ˜i−∑i=1nΔ˜iTεi2zi+∑i=1n−1Δ˜iTεizi+1.
Based on Young’s inequality, the following inequalities can be obtained:(23)2ZTPKx1≤ℏmZTPKKTPTZ+e12+ς02,2ZTPλ≤ZTPPTZ+∑i=1nλ¯i22ZTPL−1Φ∗TΥx¯,x¯^≤ZTPPTZ+∑i=1nβi2Li−12Φi∗22ZTPL−1Φ˜TMx¯^≤r1ZTPPTZ+1r1∑i=1nδi2Li−12Φ˜i22ZTPΔ˜≤ZTPPTZ+∑i=1nΔ˜iTΔ˜i−2ZTPhy≤ZTPhhTPTZ+e12+ς02∑i=1nΔ˜iTd˙i≤12∑i=1nΔ˜iTΔ˜i+12∑i=1n〉i2∑i=1nΔ˜iTεiLi−1Φi∗TΥix¯i,x¯^i≤12∑i=1nΔ˜iTΔ˜i+12∑i=1nε¯2Li−12βi2Φi∗2∑i=1nΔ˜iTεiLi−1Φ˜iTMix¯^i≤r22∑i=1nΔ˜iTΔ˜i+12r2∑i=1nε¯2Li−12δi2Φ˜i2,∑i=1nΔ˜iTεiλi≤12∑i=1nΔ˜iTΔ˜i+12∑i=1nε¯2λ¯i2,−∑i=1nΔ˜iTεi2zi≤12∑i=1nΔ˜iTΔ˜i+ε¯42ZTZ,∑i=1n−1Δ˜iTεizi+1≤12∑i=1nΔ˜iTΔ˜i+ε¯22ZTZ,where r1 and r2 are the designed positive constants, yd≤ς0, ℏm=1/Bl2ϑm2, ε¯=maxεi, λi≤λ¯i, Υix¯i,x¯^i≤βi,Mix¯^i≤δi, e1 is the tracking error, which will be given subsequently.
Substituting the above inequalities into (22) produces(24)V˙0≤−ZTΛ−Λ1−ε¯2+ε¯42I1Z+2e12−∑i=1nΔ˜iTεi−7+r22I1Δ˜i+R0+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2,where Λ1=PℏmKKT+hhT+r1+3I1PT, R0=∑i=1n1+1/2ε¯2βi2Li−12Φi∗2+2ς02+∑i=1n1+1/2ε¯2λ¯i2+1/2∑i=1nϱi2.


## 3.2. Robust Adaptive Neural Output Feedback Control Design

Step 1.
Define the following tracking errors:(25)e1=y−yd,ei=x^i−xid,i=2,…,n,where xid is the designed virtual control law.
Considering (7) and taking the time derivative of e1 yield(26)e˙1=X˙1−y˙d=ℏx˙1−y˙d=F1y+ℏx2d+e2+z2+d1−y˙d.
Because of the unknown smooth functionF1y, the following RBFNN is used to approximate it:(27)F1y=QF−1ΦF∗TMFy+σF∗y,where QF=QFT>0 is the designed matrix, MFy is Gaussian function, and ΦF∗ is the weight matrix.
Substituting (27) into (26) follows(28)e˙1=QF−1ΦF∗TMFy+QF−1σF∗y+ℏx2d+ℏe2+ℏz2+ℏd1−y˙d
Consider the Lyapunov function candidate as(29)V1=V0+12e1Te1+12trΦ˜1TΘ1−1Φ˜1+12trΦ˜FTΩF−1Φ˜F,where Θ1=Θ1T>0 and ΩF=ΩFT>0 are the designed matrixes, Φ˜F=ΦF∗−Φ^F.
Invoking (24) and (28), we obtain(30)V˙1≤−ZTΛ−Λ1−ε¯2+ε¯42I1Z+2e12+R0+e1TQF−1ΦF∗TMFy−∑i=1nΔ˜iTεi−7+r22I1Δ˜i+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2+e1TQF−1σF∗y+e1Tℏx2d+e1Tℏe2+e1Tℏz2+e1Tℏd1−e1Ty˙d−trΦ˜1TΘ1−1Φ^˙1−trΦ˜FTΩF−1Φ^˙F≤−ZTΛ−Λ1−ε¯2+ε¯42I1−CZ+e1Tℏx2d+e1TQF−1ΦF∗TMFy−∑i=1nΔ˜iTεi−7+r22I1Δ˜i+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2+5+3Bu22e1Te1+12σ¯F2QF−12+12e2Te2+12d¯12−trΦ˜1TΘ1−1Φ^˙1−trΦ˜FTΩF−1Φ^˙F−e1Ty˙d+R0,where C=diag0n×n,1/2I1,…,0n×n, σF∗≤σ¯F.
Design the virtual control functionx2d and parameter adaptive laws as(31)x2d=Hτμ,(32)τ˙i=e1iμi,i=1,2,…,n,(33)μ=q1e1−y˙d+QF−1Φ^FTMFy−d^1,(34)Φ^˙1=Θ1M1x¯^1e1TL1−1−ϖ1Φ^1,(35)Φ^˙F=ΩFMFye1TQF−1−ϖFΦ^F,where τ=τ1,τ2,…,τnT, μ=μ1,μ2,…,μnT, e1i is the ith element of e1, Hτ=diagHτ1,Hτ2,…,Hτn, Hτi is Nussbaum-type function, which is chosen as Hτi=τi2cosτi, q1=diagq11,q12,…,q1n, q1i, ϖ1 and ϖF are designed positive constants.
Considering (31)–(35), the following facts can be obtained:(36)e1Tℏx2d=∑i=1nℏiHτiτ˙i=∑i=1nℏiHτi+1τ˙i−e1Tμ,(37)e1TQF−1Φ˜FTMFy−trΦ˜FTΩF−1Φ^˙F=trΦ˜FTϖFΦ^F≤−12ϖFΦ˜F2+12ϖFΦF∗2,(38)−trΦ˜1TΘ1−1Φ^˙1=−trΦ˜1TM1x¯^1e1TL1−1−Φ˜1Tϖ1Φ^1≤e1Te1−12ϖ1−δ124L1−1Φ˜12+12ϖ1Φ1∗2,where ℏ=diagℏ1,ℏ2,…,ℏn with ℏi being the diagonal element.
Substituting (36)–(38) into (30) gives(39)V˙1≤−ZTAZ−e1Tϵ1e1−12ϖ1−δ124L1−12Φ˜12−12ϖFΦ˜F2−∑i=2nΔ˜iTεi−7+r22I1Δ˜i+12e2Te2−Δ˜1Tε1−8+r22I1Δ˜1+∑i=1nℏiHτi+1τ˙i+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2+R1,where A=Λ−Λ1−ε¯2+ε¯4/2I1−C−C1, C1=diag1/2I1,0n×n,…,0n×n, ϵ1=q1−9+3Bu2+ε¯2/2I1, R1=R0+1/2ϖFΦF∗2+1/2σ¯F2QF−12+1/2ϖ1Φ1∗2+d¯12.Step 2.
Considering (10) and differentiating e2 give(40)e˙2=x^3+f^2x¯^2−k2x^1+χ2y+d^2−x˙2d=x3d+e3+L2−1Φ^2TM2x¯^2−k2x^1+χ2y+d^2−x˙2d.
In particular, the dynamic surface control technique is employed to avoid the repeated computation ofx˙2d and achieve its available derivative. Let x2d pass the first-order filter η2 [7].(41)h2η˙2+η2=x2d,η20=x2d0,where h2=diagh21,h22,…,h2n>0 is the time constant matrix of the filter.
By definingω2=η2−x2d, we have(42)ω˙2=η˙2−x˙2d=−h2−1ω2+−∂x2d∂ττ˙−∂x2d∂e1e˙1−∂x2d∂ydy˙d−∂x2d∂Φ^FΦ^˙F−∂x2d∂yy˙−∂x2d∂d^1d^˙1,=−h2−1ω2+l2e1,e2,z2,y,y˙d,y¨d,Φ^F,d^1,where l2• is smooth function vector in regard to Π2•. Since the set Π2• is compact, the smooth function l2• has a maximum l¯2 on set Π2• for the given initial conditions.
Considering (41), the virtual control law x3d is designed as(43)x3d=−q2e2+η˙2+k2x^1−χ2y−d^2−L2−1Φ^2TM2x¯^2,where q2=diagq21,q22,…,q2n, q2i is the designed positive constant.
The parameter adaptive law is proposed as(44)Φ^˙2=Θ2M2x¯^2e2TL2−1−ϖ2Φ^2,where Θ2=Θ2T>0 is the designed matrix, and ϖ2>0 is the designed constant.
Invoking (43), we obtain(45)e˙2=−q2e2+e3−h2−1ω2+l2•.
Choose the following Lyapunov function candidate as(46)V2=V1+12e2Te2+12trΦ˜2TΘ2−1Φ˜2+12ω2Tω2.
The time derivative ofV2 is(47)V˙2=V˙1+e2Te˙2−trΦ˜2TΘ2−1Φ^˙2+ω2Tω˙2.
Considering (44), we obtain(48)−trΦ˜2TΘ2−1Φ^˙2=−trΦ˜2TM2x¯^2e2TL2−1−Φ˜2Tϖ2Φ^2≤e2Te2−12ϖ2−δ224L2−12Φ˜22+12ϖ2Φ2∗2.
Substituting (39), (45) and (48) into (47), we have(49)V˙2≤−ZTAZ−∑i=12eiTϵiei−∑i=1212ϖi−δi24Li−12Φ˜i2−12ϖFΦ˜F2−∑i=2nΔ˜iTεi−7+r22I1Δ˜i+e2Te3−Δ˜1Tε1−8+r22I1Δ˜1+∑i=1nℏiHτi+1τ˙i+R2+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2−ω2Th2−1−Iω2,where R2=R1+1/2ϖ2Φ2∗2+l¯22, ϵ2=q2−2I−1/2h2−12I.Step 3.
3≤i≤n−1: Considering (10) and differentiating ei give(50)e˙i=xi+1d+ei+1+Li−1Φ^iTMix¯^i−kix^1+χiy+d^i−x˙id.
Similar to Step2, let xid pass the following first-order filter ηi [7]:(51)hiη˙i+ηi=xid,ηi0=xid0,where hi=diaghi1,hi2,…,hin>0 is the time constant of the filter.
Defineωi=ηi−xid. Differentiating ωi yields(52)ω˙i=−hi−1ωi+−∂xid∂ei−1e˙i−1−∂xid∂ei−2e˙i−2−∂xid∂x^i−1x^˙i−1−∂xid∂Φ^i−1Φ^˙i−1−∂xid∂yy˙−∂xid∂d^i−1d^˙i−1=−hi−1ωi+liei,ei−1,y,x¯^i−1,Φ^i−1,d^i−1,where li• is a smooth function vector in regard to Πi•. Since the set Πi• is compact, the smooth function li• has a maximum l¯i on set Πi• for the given initial conditions.
Considering (51), the virtual control law is developed as(53)xi+1d=−qiei−ei−1+η˙i+kix^1−χiy−d^i−Li−1Φ^iTMix¯^i,where qi=diagqi1,qi2,…,qin, qij,j=1,2,…,n is the designed positive constant.
The parameter adaptive law is designed as(54)Φ^˙i=ΘiMix¯^ieiTLi−1−ϖiΦ^i,where Θi=ΘiT>0 is the designed matrix, and ϖi>0 is the designed constant.
Invoking (53), we obtain(55)e˙i=−qiei−ei−1+ei+1−hi−1ωi+li•.
Select the Lyapunov function candidate as(56)Vi=Vi−1+12eiTei+12trΦ˜iTΘi−1Φ˜i+12ωiTωi,
By invoking (54), it outputs(57)−trΦ˜iTΘi−1Φ^˙i=−trΦ˜iTMix¯^ieiTLi−1−Φ˜iTϖiΦ^i≤eiTei−12ϖi−δi24Li−12Φ˜i2+12ϖiΦi∗2.
Differentiating (56), we obtain(58)V˙i≤−ZTAZ−∑i=1n−1eiTϵiei−∑i=1n−112ϖi−δi24Li−12Φ˜i2−12ϖFΦ˜F2−∑i=2nΔ˜iTεi−7+r22I1Δ˜i+eiTei+1−Δ˜1Tε1−8+r22I1Δ˜1+∑i=1nℏiHτi+1τ˙i+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2+Ri−∑i=2n−1ωiThi−1−I1ωi,where Ri=Ri−1+1/2ϖiΦi∗2+l¯i2, ϵi=qi−3/2I1−1/2hi−12I1.Step 4.
Taking the derivative ofen yields(59)e˙n=u+Ln−1Φ^nTMnx¯^n−knx^1+χny+d^n−x˙nd.
Similarly, letxnd pass the following filter ηn [7]:(60)hnη˙n+ηn=xnd,ηn0=xnd0.where hn=diaghn1,hn2,…,hnn>0 is the time constant of the filter.
Defineωn=ηn−xnd. Taking the derivative of ωn yields(61)ω˙n=−hn−1ωn−∂xnd∂en−1e˙n−1−∂xnd∂en−2e˙n−2−∂xnd∂x^n−1x^˙n−1−∂xnd∂Φ^n−1Φ^˙n−1−∂xnd∂yy˙−∂xnd∂d^n−1d^˙n−1=−hn−1ωn+lnen,en−1,y,x¯^n−1,Φ^n−1,d^n−1,where ln• is smooth function vector in regard to Πn•. Since the set Πn• is compact, the smooth function ln• has a maximum l¯n on set Πn• for the given initial conditions.
Design the actual control input and parameter adaptive law as(62)u=−qnen−en−1+η˙n+knx^1−χny−d^n−Ln−1Φ^nTMnx¯^n,(63)Φ^˙n=ΘnMnx¯^nenTLn−1−ϖnΦ^n,where qn=diagqn1,qn2,…,qnn with qnjj=1,2,…,n being positive constant, Θn=ΘnT>0 is the designed matrix, and ϖn>0 is the designed constant.
Define the Lyapunov function candidate as(64)Vn=Vn−1+12enTen+12trΦ˜nTΘn−1Φ˜n+12ωnTωn.
DifferentiatingVn outputs(65)V˙n=V˙n−1+enTe˙n−trΦ˜nTΘn−1Φ^˙n+ωnTω˙n.
Considering (60)–(63), we obtain(66)e˙n=−qnen−en−1−hn−1ωn+ln•,(67)−trΦ˜nTΘn−1Φ^˙n≤enTen+12ϖnΦn∗2−12ϖn−δn24Ln−12Φ˜n2,where ρn is the designed constant.
Substituting (66) and (67) into (65), we have(68)V˙n≤−ZTAZ−∑i=1neiTϵiei−∑i=1n12ϖi−δi24Li−12Φ˜i2−12ϖFΦ˜F2−∑i=2nΔ˜iTεi−7+r22I1Δ˜i+Rn−Δ˜1Tε1−8+r22I1Δ˜1+∑i=1nℏiHτi+1τ˙i+∑i=1n1r1+ε¯22r2Li−12δi2Φ˜i2−∑i=2nωiThi−1−I1ωi≤−ZTAZ−∑i=1nϵieiTei−Δ˜1Tε1−8+r22I1Δ˜1−12ϖFΦ˜F2−∑i=2nΔ˜iTεi−7+r22I1Δ˜i+Rn−∑i=1n12ϖi−14+1r1+ε¯22r2δi2Li−12Φ˜i2−∑i=2nωiThi−1−I1ωi+∑i=1nℏiHτi+1τ˙i,where Rn=Rn−1+1/2ϖnΦn∗2+l¯n2, ϵn=qn−3/2I1−1/2hn−12I1.


## 3.3. Stability Analysis

Hereto, the following theorem is proposed to reveal the main results of this work.Theorem 1.
Consider a class of MIMO nonlinear systems (1) satisfying assumption 1–3. The state observer is designed as (10), and the NDOs are developed as (16-17), and the virtual control laws are proposed as (31), (43), and (53). Under the presented output feedback FTC scheme (62), all tracking error signals of the closed-loop system are uniformly ultimately bounded.Proof.
From (68), it is observed that(69)V˙n≤−sVn+Γ+∑i=1nℏiHτi+1τ˙i,where s and Γ are given by(70)s=minλminAλmaxP,2λminϵi,λmin2ε1−8+r2I1,ϖFλmaxΩF−1,λmin2ε1−7+r2I1,2λminhi−1−I1,ϖi−1/2+2/r1+ε¯2/r2δi2Li−12λmaxΘi−1,Γ=∑i=1n1+12ε¯2βi2Li−12Φi∗2+∑i=1n1+12ε¯2λ¯i2+12ϖFΦF∗2+12σ¯F2QF−12+d¯12+2ς02+∑i=1n12ϖiΦi∗2+∑i=2nl¯i2+12∑i=1nϱi2.
According to Lemma1, it can be obtained that Vn and ∑i=1nℏiHτi+1τ˙i are bounded over 0,tf. By defining Γ1=maxt∈0,tf∑i=1nℏiHτi+1τ˙i, we obtain(71)V˙n≤−sVn+Γ¯,where Γ¯=Γ+Γ1.
Integrating (71) over 0,t yields(72)Vn≤Γ¯s+Vn0−Γ¯se−st.
From (72) and the definition of Vn, we obtain that Vn is exponentially convergent, i.e., limt⟶∞Vn=Γ¯/s. Hence, all the error signals of the closed-loop remain bounded. This concludes the proof.Remark 4.
In this paper, the actuator faults, which are coupled with the unknown control gains, are considered, and the Nussbaum function is utilized to eliminate their adverse effect. Compared with some direct adaptive method [11], the developed control algorithm can effectively avoid the problem of singularity and simplify the proof process. Compared with some existing observer-based output feedback control methods [18, 20, 43], the proposed robust FTC scheme can guarantee the closed-loop stability for the MIMO nonlinear system with multiple negative effects simultaneously, including unmeasured states, unknown nonlinearity, unknown control directions, actuator faults, and external disturbances.Remark 5.
In the control design, different design parameters will result in different system performance. Choosing largerϵi will increase the control gains and is beneficial for tracking results within bounds. However, excessive control gains require more control energy and may cause input saturation. Consequently, in order to achieve applicable control performance, the selection of corresponding design parameters should satisfy the following conditions and reach a compromise:(73)A>0,ϵi>0,2ε1−8+r2I1>0,hi−1−I1>0,ϖF>0,ϖi−12+2r1+ε¯2r2δi2Li−12>0.


## 4. Simulation Examples

In this section, simulations studies on the unmanned helicopter position movement are given to illustrate the feasibility of the obtained results. The position dynamic model of unmanned helicopter with actuator faults and external disturbances can be described as [44](74)P˙=V,mV˙=BϑFR+mgξ+Dt,y=P,where g is the gravitational acceleration, ξ=0,0,1T, and m defines the gross mass of the unmanned helicopter. P=X,Y,ZT and V=u,v,wT denote the position and velocity vectors, respectively. B=diagB1,B2,B3 is the unknown constant control gain matrix, ϑ=diagϑ1,ϑ2,ϑ3, ϑi is the remaining control efficiency, Dt is external time-varying unknown disturbance, FR=FR, F=Fx,Fy,FzT is the control input, and R refers to the rotation matrix whose definition can be found in [44].In the simulation, we suppose that the unmanned helicopter is operated in constant attitude angles. Furthermore, the initial positions are assumed asX0,Y0,Z0T=1,1,0T m. Here, the unmanned helicopter is required to track the following desired trajectories(75)Xd=10cosπ10tm,Yd=10sinπ10tm,Zd=−101−e−3tm,.The external time-varying disturbances are given by(76)D=10sin0.2t1.2sin0.1t15sin0.5t.In view of the analysis process in Section3, the control object is to design control input vector F to make the output y track the desired trajectory yd as closely as possible. The basic parameters of the unmanned helicopter are selected as m=800 kg, g=9.8, ϕ=10°, θ=15° and ψ=20°. The control gains and actuator faults are assumed as B=diag1.1,1.2,1.3 and ϑ=diag0.8,0.7,0.6, respectively. The relevant designed parameters are selected as q1=diag5,5,5, q2=diag1000,200,500, k1=diag2,2,2, k2=diag1,1,1, χ1=diag2,2,2, χ2=diag1,1,1, ϖ1=0.001, L1=diag20,20,20.First, the position tracking errore1=y−yd without robust FTC is shown in Figure 2. It can be noted that all tracking errors deviate from the origin severely. To reflect the movement of the helicopter more visually, Figure 3 is presented, where the magenta lines P d represent the desired trajectories, and the black lines P s are the actual outputs. From Figure 3, it can be seen that if there is no effective robust FTC scheme, satisfactory trajectory tracking performance cannot be obtained. Simulation results of Figures 2 and 3 reveal that the actuator faults, which are coupled with the unknown control directions and external disturbances, have a great negative influence on the control performance of the helicopter system. If these negative effects can not be tackled timely, they will lead to degradation of system performance and even catastrophic consequences.Figure 2
Tracking errorse1i without robust FTC.Figure 3
Tracking trajectoriesy and yd without robust FTC.In order to ensure the safe flight for the unmanned helicopter with unavailable states, unknown control directions, actuator faults, and external disturbances, a robust adaptive NN output feedback FTC scheme is developed, and the corresponding simulations are given by Figures4–7. Figure 4 provides the contrastive results of tracking errors, where the blue lines e1ip denote the tracking errors under the proposed controller, and the red lines e1in denote these under adaptive FTC strategy proposed in [11]. According to Figure 4, it can be observed that both control schemes can guarantee that the tracking errors converge to a small boundary. However, compared with the adaptive FTC strategy proposed in [11], the developed control algorithm has faster convergence rate and smaller tracking error. Meanwhile, the three-dimensional graph of the output trajectory y is provided in Figure 5, where the blue lines P p denote the system output under the proposed controller, and the red lines P n refer to these under adaptive FTC strategy proposed in [11]. From Figure 5, we can see that the presented control scheme can make the system output reach the desired trajectory faster, which reflects the same conclusion as Figure 4. The curve of designed fault tolerant controller is provided in Figure 6. Figure 6 manifests that the control command signals are convergent and can adjust dynamically to restrain the adverse impacts. Furthermore, the RBFNN weight matrix changes in reasonable boundary as time goes on, which is displayed in Figure 7.Figure 4
Contrastive results of tracking errors under proposed controller and adaptive FTC method [11].Figure 5
Contrastive results of outputy under the proposed controller and the adaptive FTC method [11].Figure 6
Input signals of the proposed controller.Figure 7
The norm of the RBFNN weight matrix estimationM^.


## 5. Conclusion

In this paper, a NN-based adaptive output feedback FTC scheme has been proposed for uncertain MIMO nonlinear systems with unmeasured states, unknown control directions, actuator faults, and external disturbances. The RBFNN has been utilized to approximate the unknown nonlinear function, and the NDO has been constructed to estimate the external disturbances. Especially, the unknown actuator faults and control directions have been handled by the constructed Nussbaum function. The stability of the closed-loop system has been proved, and the performance of the presented controller has been confirmed through simulation results on unmanned helicopter position movement. In the future, the novel robust FTC strategy for the variable gain nonlinear systems with random faults and disturbances is worthy of further exploration, which are the other common difficulties in practice.



---

*Source: 4770439-2022-05-29.xml*