# Adaptive Constrained Control for Uncertain Nonlinear Time-Delay System with Application to Unmanned Helicopter

Authors: Rong Li; Qingxian Wu; Qingyun Yang; Hui Ye
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper investigates a class of nonlinear time-delayed systems with output prescribed performance constraint. The neural network and DOB (disturbance observer) are designed to tackle the uncertainties and external disturbance, and prescribed performance function is constructed for the output prescribed performance constrained problem. Then the robust controller is designed by using adaptive backstepping method, and the stability analysis is considered by using Lyapunov-Krasovskii. Furthermore, the proposed method is employed into the unmanned helicopter system with time-delay aerodynamic uncertainty. Finally, the simulation results illustrate that the proposed robust prescribed performance control system achieved a good control performance.


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## Body

## 1. Introduction

Time-delay systems have drawn considerable attention in the past decade [1, 2]. The adaptive backstepping technology was employed into the uncertain nonlinear time-delay system in [3]. The dynamic surface method was presented for the nonlinear time-delay system in [4]. In [5], the nonlinear stochastic system with time delay was studied. The finite-time control method was proposed for a class of time-delay systems in [6, 7]. In the previous studies on time-delay system, the uncertain nonlinear systems consisting of both constraint and external disturbances were not considered. In this paper, we will study a class of uncertain nonlinear time-delay systems subject to constraint.It is well known that the uncertainty and external disturbance have an effect on the tracking performance of closed systems. Neural network is popular for its ability to cope with uncertainty [8]. In [9], the neural network was introduced into a class of nonlinear systems with unknown coefficient matrices. Combining RBFNN (radial basis function neural network) and disturbance observer, fault tolerant control method was presented to deal with input saturated system with actuator faults in [10]. Moreover, the disturbance observer is a valid method to deal with external disturbance [11]. In [12], the disturbance observer was proposed for permanent-magnet synchronous motor drivers. In [13], the sliding mode disturbance observer was presented to deal with mismatched disturbance. In [14], the disturbance observer was employed into a transport aircraft control system subject to continuous heavy cargo airdrop. In this paper, the neural network and disturbance observer will be utilized to tackle uncertainties, time delay, and external disturbance.Another challenging problem in controller design lies in the constrained condition of the nonlinear systems [15]. The existence of constraint condition may degrade the performance or cause the instability of the closed control systems [16]. Using the Barrier Lyapunov function and adaptive backstepping technology, a robust constrained controller for a class of nonlinear strict systems was presented in [17]. In [18], the Barrier Lyapunov function was employed into the switched systems subject to output constraints. In [19], the Barrier Lyapunov function and high-gain observer were introduced to deal with the constrained trajectory tracking problem of the marine surface vessel. Additionally, prescribed performance is another method to cope with output constraints, by defining the appropriate prescribed performance. In [20], the prescribed performance-based feedback linearization method was proposed to deal with output tracking error constraints for the MIMO (multiple-input multiple-output) nonlinear systems. In [21], the prescribed performance and adaptive fuzzy logic were employed into the nonlinear adaptive controller design. To the best of the authors’ knowledge, there is still no research about uncertain nonlinear time-delay system considering uncertainties, external disturbance, and output constraints. Thus, in this paper, we will present a prescribed performance-based adaptive constrained control method for the time-delay nonlinear systems.Nowadays, the unmanned helicopter system has received an increasing attention, and there is an amount of studies about the flight control approaches [22, 23]. In [24], chattering-free sliding mode was proposed for the miniature helicopter system. To solve the tracking problem with nonlinearity, the model predictive control method for unmanned helicopters was presented in [25]. In [26], a trajectory tracking control method was proposed for unmanned helicopter system with constraint conditions. However, with the increasing demands for real time and accuracy, the aerodynamic disturbance caused by transmission delay for unmanned helicopter control system cannot be ignored. In this paper, we will apply the prescribed performance-based robust adaptive control approach for the uncertain unmanned helicopter systems with external disturbance, time delay, and output constraints.This paper is organized as follows. In Section2, problem statement and preliminaries of time-delay system and prescribed performance are introduced. Section 3 presents the entire adaptive controller design and stability analysis. In Section 4, the prescribed performance-based control method is employed into the unmanned helicopter system. Finally, simulation and conclusion are given in Sections 5 and 6, respectively.


## 2. Problem Statement and Preliminaries

In this subsection, we will review some preliminary knowledge about nonlinear time-delay system, prescribed performance, and neural network, which are necessary in the following controller design. Firstly, consider a class of uncertain MIMO nonlinear time-delay systems in the form of(1)x˙it=fix-it+gix-itxi+1t+hix-it-τi+Δfix-it+dit,1≤i≤n-1x˙nt=fnx-nt+gnx-ntut+hnx-nt-τn+Δfnx-nt+dntyt=x1t,where xit∈Rm, i=1,2,…,n, are the system state vectors which are assumed to be measurable, x-it=x1Tt,x2Tt,…,xiTtT∈Rm·i, y∈Rm is the output vector, and ut∈Rm is the control input vector. fix-it∈Rm, i=1,2,…,n, are the known smooth nonlinear function, gix-i∈Rm×m, i=1,2,…,n, represent the known control coefficient matrices, and hix-it-τi∈Rm×m, i=1,2,…,n, indicate the known time-delay functions. Δfix-it∈Rm, i=1,2,…,n, denote the unknown nonlinear functions which contain both parametric and nonparametric uncertainties. τi∈R, i=1,2,…,n, stand for constant unknown time delays. dit∈Rm, i=1,2,…,n, mean the unknown external disturbance. In this paper, we impose the following assumptions and lemmas.Assumption 1 (see [5]).
The ideal tracking signalsyrt and their derivatives y˙rt are known and continuous.Assumption 2 (see [11]).
The external disturbancedit means the slow varying signal, while it is restricted in the bound of d˙it≤d-i, d-i>0.Assumption 3 (see [10]).
Fori=1,…,n, the known continuous function gix-it satisfies gix-it≠0.Lemma 4 (see [9]).
Consider a class of nonlinear systemsx˙=fx. For any initial conditions x0∈Rn, if there exists a C1 continuous and positive definite Lyapunov function satisfying ϖ1x≤Vx≤ϖ2x, ϖix:Rn→R, i=1,2, mean the K class functions, such that V˙≤-c1Vx+c2, c1>0, c2>0, then one can conclude that the solution xt is uniformly bounded.Lemma 5 (see [3]).
Forv>0, if there exists one set Z defined by Z≔z∣z≤0.8814v, then one can conclude that, for any z∉Z, the inequality 1-2tanh2⁡z/v<0 is satisfied.Lemma 6 (see [10]).
For any continuous functionfz, there exists a valid linear neural network to approximate it by choosing enough nodes on the compact set z∈Nz. The basic function can be selected as Gaussian function siz=e-z-μi2/ci, where ci>0 is the width of function and μi∈R represent the center of function. The neural network approximator consists of the weight estimate vector W^z and Gaussian function Sz=s1z,…,snzT, which can be written as fpz=W^TzSz+ε. Assume that W∗z means the optimal approximating weight, such that (2)W∗z≔arg⁡minW^∈Rn⁡supz∈Ωz⁡fz-fpz.In addition, the optimal approximator of continuous function fz can be written as (3)fz=W∗TzSz+ε∗,where ε∗ indicates the optimal approximate error.


## 3. Controller Design and Stability Analysis

In this section, the objective is to propose a robust prescribed performance control law for uncertain nonlinear systems such that the closed-loop errors converge to a small neighborhood of the origin.


### 3.1. Prescribed Performance Controller Design

Step 1.
Define the tracking errorsz~1t∈Rm and z2t∈Rm as follows: (4)z~1t=x1t-ydtz2t=x2t-x2∗t,where ydt∈Rm is the ideal tracking signal and x2∗t∈Rm is the immediate control. The output error transformation can be defined in the form of [18] (5)z1jt=Q-1z~1jtρjt,α-jt,α_jt=12ln⁡α_j+z~1jt/ρjtα-j-z~1jt/ρjt,j=1,…,m,where α-j and α_j are the positive constants and ρjt indicates the performance function, which can be chosen as [26] (6)ρjt=ρj0-ρj∞e-lt+ρj∞.The constant ρj∞>0 is the maximum amplitude of the tracking error at the steady state. The decreasing rate e-ljt of ρjt represents the desired convergence speed of the tracking error. Therefore, the appropriate choice of the performance function ρjt and the design constant imposes bounds on the system output trajectory.
Definef1=f11,…,f1mT∈Rm and g1=g11T,…,g1mTT∈Rm×m. Thus, the time derivative of z1jt becomes (7)z˙1jt=∂z1jt∂z~1jt/ρjt1ρjtz~˙1jt-∂z1jt∂z~1jt/ρjtz~1jtρj2tρ˙jt=12α¯j+α_jα_j+z~1jt/ρjtα¯j-z~1jt/ρjt1ρjtz~˙1jt-z~1jtρj2tρ˙jt=12α¯j+α_jα_j+z~1jt/ρjtα¯j-z~1jt/ρjt·-z~1jtρj2tρ˙jt+f1jx1t+g1jx1tx2t+h1jx1t-τ1+Δf1jx1t+d1jt-y˙djtρjtIn order to simplify the analysis, we define M1jand N1jas follows: (8)M1jt=α-j+α_j2ρjα_j+z~1jt/ρjtα-j-z~1jt/ρjt>0.N1jt=α-j+α_j2ρj2tα_j+z~1jt/ρjtα-j-z~1jt/ρjtz~1jtρ˙jtFurthermore, we can define M1t=diag⁡M11t,…,M1mt and N1t=N11t,…,N1mtT, then we have (9)z˙1t=M1tz~˙1t-N1t=M1tf1x1t+g1x1tx2t-y˙dt+Δf1x1t+h1x1t-τ1+d1t-N1t.Since Δf1x1t is unknown, using the RBFNN to approximate it, we obtain (10)M1tΔf1x1t+γ1t=η1-1W1∗TS1+ε1∗,where γ1t=γ1,1t,…,γ1,mtT, and (11)γ1,jt=16z~1jttanh2⁡z~1jtv1h1j2x1t,j=1,…,m.Substituting (10) into (1) results in(12)x˙1t=f1x1t+g1x1tx2t+h1x1t-τ1-M1-1tγ1t+η1-1M1-1tW1∗TS1+χ1t.Invoking (4), (9), and (10), the time derivative of z1t can be rewritten as (13)z˙1t=M1tf1x1t+g1x1tz2t+g1x1tx2∗t+h1x1t-τ1-y˙dt-γ1t+η1-1W1∗TS1+M1tχ1t-N1t.Construct updating law of the RBFNN (14)W^˙1=Γ1η1-1z1tS1-σ1Γ1-1W^1,where σ1>0 is a design parameter. Furthermore, the DOB can be chosen as (15)χ^1t=w1t+η1x1tw˙1t=-η1w1t-η1f1x1t+g1x1tx2t+η1-1W^1TS1+η1x1t+M1Ttz1t.According to (15), we obtain (16)χ^˙1t=η1χ~1t+M1Ttz1t+W~1TS1.According to the neural network updating law (14) and DOB (15), the immediate control is chosen (17)x2∗t=g1-1x1t-M1-1tK1z1t-f1x1t+y˙dt+M1-1tN1t-M1-1tγ1t-η1-1W^1TS1-χ^1t,where M1t=diag⁡M11t,…,M1mt. y˙dt is the time derivative of reference trajectory, and K1∈Rm×m is the constant positive definite matrices. W^1 is the estimated values of W1∗, W~1 represents the estimated error, and W~1=W1∗-W^1. Substituting (17) into (13), the time derivative of z1t becomes (18)z˙1t=-K1z1t+M1tg1x1tz2t-γt+M1th1x1t-τ1+η1-1W~1TS1+M1tχ~1t.Choose the Lyapunov-Krasovskii functional candidate as (19)V1=z1Ttz1t2+∫t-τ1tP1x1τdτ+trW~1TΓ1-1W~1+12χ~1Ttχ~1t,where the positive function P1x1t can be designed as follows: (20)P1x1t=h1Tx1th1x1t2.Then the time derivative of V1 is (21)V˙1=z1Tt-K1z1t+M1tg1x1tz2t+M1th1x1t-τ1-γ1t+η1-1W~1TS1+M1tχ1t+P1x1t-P1x1t-τ1-trW~1TΓ1-1W^˙1-χ~1Ttχ^˙1t+χ~1Ttχ˙1t=-z1TtK1z1t+z1TtW~1TS1+z1Tth1x1t-τ1-z1Ttγ1t+z1TtM1tg1x1tz2t+η1-1z1TtW~1TS1+z1TtM1tχ~1t+P1x1t-P1x1t-τ1-trW~1TΓ1-1W^˙1-χ~1Ttχ^˙1t+χ~1Ttχ˙1t.Substituting (14) into (21), we obtain (22)V˙1=-z1TtK1z1t+z1TtW~1TS1+z1Tth1x1t-τ1+z1TtM1tg1x1tz2t-z1Ttγ1t+z1TtM1tχ~1+P1x1t-P1x1t-τ1+σ1trW~1TΓ1-1W^1-χ~1Ttχ^˙1t+χ~1Ttχ˙1t.Substituting (16) into (22) yields (23)V˙1=-z1TtK1z1t+z1TtW~1TS1+z1Tth1x1t-τ1-z1Ttγ1t+z1TtM1tg1x1tz2t+P1x1t-P1x1t-τ1+σ1trW~1TΓ1-1W^1-η1χ~1Ttχ~1t-χ~1TtW~1TS1+χ~1Ttχ˙1t.Step i. Define the error variables zit∈Rm and zi+1t∈Rm(24)zit=xit-xi∗tzi+1t=xi+1t-xi+1∗t,where xi∗t is a virtual control law. Combining (1) and (24) and differentiating zit with respect to time, we have (25)z˙it=fix-it+gix-itxi+1t+hix-it-τi+Δfix-it+dit-x˙i∗t.Since Δfix-it is unknown, we use RBFNN to approximate uncertain function. W^i indicates the estimate of Wi∗, and εi∗ is the approximate error. W~i represents the estimated error, which is defined as W~i=Wi∗-W^i. Define the variable χit=εi∗+dit. (26)Δfix-it+γit=ηi-1Wi∗TSi+εi∗,where γit=γi,1t,…,γi,mtT, and (27)γi,jt=16zijttanh2⁡zijtvihij2x-it,j=1,…,m.Substituting (26) into (1) yields (28)x˙it=fix-it+gix-itxi+1t+hix-it-τi-γit+ηi-1Wi∗TSi+χit.Moreover, substituting (26) into (25), we have (29)z˙it=fix-it+gix-itxi+1t+hix-it-τi-γit+ηi-1Wi∗TSi+χit-x˙i∗t.Construct updating law of the RBFNN (30)W^˙i=Γiη1-1zitSi-σiΓi-1W^i,where σi>0 is a design parameter. Furthermore, the DOB can be chosen as (31)χ^it=wit+ηixitw˙it=-ηiwit-ηifix-it+gix-itxi+1+ηi-1W^iTSi+ηixit+zit.According to (31), we obtain (32)χ^˙it=ηiχ~it+zit+W~iTSi.Hence, the virtual control law xi+1∗t is proposed as (33)xi+1∗t=gi-1x-it-Kizit-gi-1Tx-i-1tzi-1t-fix-it+x˙i∗t-ηi-1W^iTSi-χ^it.Choose the Lyapunov-Krasovskii functional candidate as (34)Vi=12ziTtzit+∫t-τitPix-iτdτ+trW~iTΓi-1W~i+12χ~iTtχ~it,where the positive function Pix-iτ is given in the form of (35)Pix-iτ=12hiTx-iτhix-iτ.Similar to Step 1, the time derivative of V˙i can be rewritten as (36)V˙i=-ziTtKizit+ziTtgix-itzi+1t-ziTtgi-1Tx-i-1tzi-1t-ziTtγit+ziTthix-it-τi+Pix-it-Pix-it-τi+σitrW~iTΓi-1W^i-ηiχ~iTtχ~it-χ~iTtW~iTSi+χ~iTtχ˙it.Step n. Define the error variable znt∈Rm: (37)znt=xnt-xn∗tInvoking (1) and (37), differentiating znt with respect to time yields (38)z˙nt=fnx-nt+gnx-ntut+hnx-nt-τ+Δfnx-nt-x˙n∗t.Since Δfnx-nt is unknown, the RBFNN is used to approximate Δfnx-nt. W^n indicates the estimate of Wn∗, and εn∗ is the approximate error. W~n represents the estimated error, which is defined as W~n=Wn∗-W^n. Define the variable χn=εn∗+dnt. (39)Δfnx-nt+γnt=ηn-1Wn∗TSn+εn∗,where γnt=γn,1t,…,γn,mtT, and (40)γn,jt=16znjttanh2⁡znjtvnhnj2x-nt,j=1,…,m.Substituting (39) into (1) yields (41)x˙nt=fnx-nt+gnx-ntut+hnx-nt-τn-γnt+ηn-1Wn∗TSn+χnt.Invoking (38) and (39), one obtains (42)z˙nt=fnx-nt+gnx-ntut+hnx-nt-τn-γnt+ηn-1Wn∗TSn+χnt-x˙n∗t.Construct updating law of the RBFNN (43)W^˙nt=Γnηn-1zntSn-σnΓn-1W^nt,where σn∈R, σn>0, is a design parameter. Furthermore, the DOB can be chosen as (44)χ^nt=wnt+ηnxntw˙nt=-ηnwnt-ηnfnx-nt+gnx-ntut+ηn-1W^nTSn+ηnxnt+znt.According to (44), we obtain (45)χ^˙nt=ηnχ~nt+znt+W~nTSn.Therefore, design the control law ut as (46)ut=gn-1x-nt-Knznt-gn-1Tx-n-1tzn-1t-fnx-nt+x˙n∗t-ηn-1W^nTSn-χ^nt,where Kn∈Rm×m, Kn>0 is design matrix. Choose the Lyapunov-Krasovskii functional candidate as (47)Vn=12znTtznt+∫t-τntPnx-nτdτ+trW~nTΓn-1W~n+12χ~nTtχ~ntand the positive function Pnx-nτ can be designed as follows: (48)Pnx-nτ=12hnTx-nτhnx-nτ.According to the derivatives in Step 1 and Step i, we have (49)V˙n=-znTtKnznt-znTtgn-1Tx-n-1tzn-1t+znTthnx-nt-τn-znTtγnt+Pnx-nt-Pnx-nt-τn-σntrW~nTΓn-1W^n-ηnχ~nTtχ~nt-χ~nTtW~nTSn+χ~nTtχ˙nt.From the above inductive design procedure, moreover, we can conclude the following theorem.


### 3.2. Stability Analysis

Theorem 7.
Considering the system dynamics described by (1), under the adaptive neural updated laws (14), (30), and (43), the disturbance observers (15), (31), and (45), and the prescribed performance-based control laws (17), (33), and (46), we can conclude that the trajectories of the closed-loop system are semiglobally uniformly bounded, while the tracking error z~1t converges to a compact set Qz asymptotically, where C and κ are defined in (50).(50)κ=min⁡λminKi-ϑξi+12,ηi-12ϑξi-δSφi2-πi2,σi2C=∑i=1nσi2λminΓi-1Wi∗2+∑i=1n12πiχ˙it2,where Qz can be made as small as desired by appropriately choosing design parameters. λmin represents the minimum eigenvalue of the matrix. λmin represents the minimum eigenvalue of the matrix.Proof.
For analytical purposes, define the total Lyapunov function(51)VΞ=∑i=1nVi,where definition of Vi can be referred to (19), (34), and (47). According to (23), (36), and (49), we obtain (52)V˙Ξ=-∑i=1nziTtKizit-∑i=1nηiχ~iTtχ~it+∑i=1nziTthix-it-τi+∑i=1nziTtχ~it-∑i=1nziTtγit-∑i=1nχ~iTtW~iTSi+∑i=1nχ~iTtχ˙it+∑i=1nPix-it-Pix-it-τi-∑i=1nσitrW~iTΓi-1W^i.Considering the fact that (53)ziTthix-it-τi≤12ziTtzit+12hiTx-it-τihix-it-τiand Pix-it-τi=1/2hiTx-it-τihix-it-τi, we have (54)V˙Ξ≤-∑i=1nziTtKizit-∑i=1nηiχ~iTtχ~it+∑i=1nziTtχ~it-∑i=1nziTtγit-∑i=1nχ~iTtW~iTSi+∑i=1nχ~iTtχ˙it+12∑i=1nziTtzit+∑i=1nPix-it-∑i=1nσitrW~iTΓi-1W^i.Then we have the following results: (55)V˙Ξ≤-∑i=1nziTtKi-12Im×mzit-∑i=1nηiχ~iTtχ~it+∑i=1nziTtχ~it-∑i=1nziTtγit-∑i=1nχ~iTtW~iTSi+∑i=1nχ~iTtχ˙it+∑i=1nPix-it-∑i=1nσitrW~iTΓi-1W^i.In addition, it is clear that there exists Si≤δSi, δSi>0; we have the following facts: (56)σitrW~iTΓi-1W^i≤-σi2trW~iTΓi-1W~i+σi2trWi∗TΓi-1Wi∗ziTtχ~it≤ϑξi2ziTtzit+12ϑξiχ~it2χ~iTtW~iTSi≤δSiφi2χ~it2+δSi2φiW~i2χ~iTtχ˙it≤πi2χ~it2+12πiχ˙it2,where ϑξi>0 and φi>0 are the constants, and then we have (57)V˙Ξ≤-∑i=1nλminKi-ϑξi+12zit-∑i=1nηi-12ϑξi-δSiφi2-πi2χ~it-∑i=1nσi2trW~iTΓi-1W~i+12∑i=1n∑j=1m1-16tanh2⁡zijtvihijTx-ithijx-it+∑i=1nσi2trWi∗TΓi-1Wi∗+∑i=1n12πiχ˙it.Invoking Lemma 5, we obtain (58)V˙Ξ≤-κVΞ+C.κ>0,C>0 are defined in (50). Therefore, according to Lemma 4, we can conclude that the solution of the closed-loop system remains within a compact subset.


## 3.1. Prescribed Performance Controller Design

Step 1.
Define the tracking errorsz~1t∈Rm and z2t∈Rm as follows: (4)z~1t=x1t-ydtz2t=x2t-x2∗t,where ydt∈Rm is the ideal tracking signal and x2∗t∈Rm is the immediate control. The output error transformation can be defined in the form of [18] (5)z1jt=Q-1z~1jtρjt,α-jt,α_jt=12ln⁡α_j+z~1jt/ρjtα-j-z~1jt/ρjt,j=1,…,m,where α-j and α_j are the positive constants and ρjt indicates the performance function, which can be chosen as [26] (6)ρjt=ρj0-ρj∞e-lt+ρj∞.The constant ρj∞>0 is the maximum amplitude of the tracking error at the steady state. The decreasing rate e-ljt of ρjt represents the desired convergence speed of the tracking error. Therefore, the appropriate choice of the performance function ρjt and the design constant imposes bounds on the system output trajectory.
Definef1=f11,…,f1mT∈Rm and g1=g11T,…,g1mTT∈Rm×m. Thus, the time derivative of z1jt becomes (7)z˙1jt=∂z1jt∂z~1jt/ρjt1ρjtz~˙1jt-∂z1jt∂z~1jt/ρjtz~1jtρj2tρ˙jt=12α¯j+α_jα_j+z~1jt/ρjtα¯j-z~1jt/ρjt1ρjtz~˙1jt-z~1jtρj2tρ˙jt=12α¯j+α_jα_j+z~1jt/ρjtα¯j-z~1jt/ρjt·-z~1jtρj2tρ˙jt+f1jx1t+g1jx1tx2t+h1jx1t-τ1+Δf1jx1t+d1jt-y˙djtρjtIn order to simplify the analysis, we define M1jand N1jas follows: (8)M1jt=α-j+α_j2ρjα_j+z~1jt/ρjtα-j-z~1jt/ρjt>0.N1jt=α-j+α_j2ρj2tα_j+z~1jt/ρjtα-j-z~1jt/ρjtz~1jtρ˙jtFurthermore, we can define M1t=diag⁡M11t,…,M1mt and N1t=N11t,…,N1mtT, then we have (9)z˙1t=M1tz~˙1t-N1t=M1tf1x1t+g1x1tx2t-y˙dt+Δf1x1t+h1x1t-τ1+d1t-N1t.Since Δf1x1t is unknown, using the RBFNN to approximate it, we obtain (10)M1tΔf1x1t+γ1t=η1-1W1∗TS1+ε1∗,where γ1t=γ1,1t,…,γ1,mtT, and (11)γ1,jt=16z~1jttanh2⁡z~1jtv1h1j2x1t,j=1,…,m.Substituting (10) into (1) results in(12)x˙1t=f1x1t+g1x1tx2t+h1x1t-τ1-M1-1tγ1t+η1-1M1-1tW1∗TS1+χ1t.Invoking (4), (9), and (10), the time derivative of z1t can be rewritten as (13)z˙1t=M1tf1x1t+g1x1tz2t+g1x1tx2∗t+h1x1t-τ1-y˙dt-γ1t+η1-1W1∗TS1+M1tχ1t-N1t.Construct updating law of the RBFNN (14)W^˙1=Γ1η1-1z1tS1-σ1Γ1-1W^1,where σ1>0 is a design parameter. Furthermore, the DOB can be chosen as (15)χ^1t=w1t+η1x1tw˙1t=-η1w1t-η1f1x1t+g1x1tx2t+η1-1W^1TS1+η1x1t+M1Ttz1t.According to (15), we obtain (16)χ^˙1t=η1χ~1t+M1Ttz1t+W~1TS1.According to the neural network updating law (14) and DOB (15), the immediate control is chosen (17)x2∗t=g1-1x1t-M1-1tK1z1t-f1x1t+y˙dt+M1-1tN1t-M1-1tγ1t-η1-1W^1TS1-χ^1t,where M1t=diag⁡M11t,…,M1mt. y˙dt is the time derivative of reference trajectory, and K1∈Rm×m is the constant positive definite matrices. W^1 is the estimated values of W1∗, W~1 represents the estimated error, and W~1=W1∗-W^1. Substituting (17) into (13), the time derivative of z1t becomes (18)z˙1t=-K1z1t+M1tg1x1tz2t-γt+M1th1x1t-τ1+η1-1W~1TS1+M1tχ~1t.Choose the Lyapunov-Krasovskii functional candidate as (19)V1=z1Ttz1t2+∫t-τ1tP1x1τdτ+trW~1TΓ1-1W~1+12χ~1Ttχ~1t,where the positive function P1x1t can be designed as follows: (20)P1x1t=h1Tx1th1x1t2.Then the time derivative of V1 is (21)V˙1=z1Tt-K1z1t+M1tg1x1tz2t+M1th1x1t-τ1-γ1t+η1-1W~1TS1+M1tχ1t+P1x1t-P1x1t-τ1-trW~1TΓ1-1W^˙1-χ~1Ttχ^˙1t+χ~1Ttχ˙1t=-z1TtK1z1t+z1TtW~1TS1+z1Tth1x1t-τ1-z1Ttγ1t+z1TtM1tg1x1tz2t+η1-1z1TtW~1TS1+z1TtM1tχ~1t+P1x1t-P1x1t-τ1-trW~1TΓ1-1W^˙1-χ~1Ttχ^˙1t+χ~1Ttχ˙1t.Substituting (14) into (21), we obtain (22)V˙1=-z1TtK1z1t+z1TtW~1TS1+z1Tth1x1t-τ1+z1TtM1tg1x1tz2t-z1Ttγ1t+z1TtM1tχ~1+P1x1t-P1x1t-τ1+σ1trW~1TΓ1-1W^1-χ~1Ttχ^˙1t+χ~1Ttχ˙1t.Substituting (16) into (22) yields (23)V˙1=-z1TtK1z1t+z1TtW~1TS1+z1Tth1x1t-τ1-z1Ttγ1t+z1TtM1tg1x1tz2t+P1x1t-P1x1t-τ1+σ1trW~1TΓ1-1W^1-η1χ~1Ttχ~1t-χ~1TtW~1TS1+χ~1Ttχ˙1t.Step i. Define the error variables zit∈Rm and zi+1t∈Rm(24)zit=xit-xi∗tzi+1t=xi+1t-xi+1∗t,where xi∗t is a virtual control law. Combining (1) and (24) and differentiating zit with respect to time, we have (25)z˙it=fix-it+gix-itxi+1t+hix-it-τi+Δfix-it+dit-x˙i∗t.Since Δfix-it is unknown, we use RBFNN to approximate uncertain function. W^i indicates the estimate of Wi∗, and εi∗ is the approximate error. W~i represents the estimated error, which is defined as W~i=Wi∗-W^i. Define the variable χit=εi∗+dit. (26)Δfix-it+γit=ηi-1Wi∗TSi+εi∗,where γit=γi,1t,…,γi,mtT, and (27)γi,jt=16zijttanh2⁡zijtvihij2x-it,j=1,…,m.Substituting (26) into (1) yields (28)x˙it=fix-it+gix-itxi+1t+hix-it-τi-γit+ηi-1Wi∗TSi+χit.Moreover, substituting (26) into (25), we have (29)z˙it=fix-it+gix-itxi+1t+hix-it-τi-γit+ηi-1Wi∗TSi+χit-x˙i∗t.Construct updating law of the RBFNN (30)W^˙i=Γiη1-1zitSi-σiΓi-1W^i,where σi>0 is a design parameter. Furthermore, the DOB can be chosen as (31)χ^it=wit+ηixitw˙it=-ηiwit-ηifix-it+gix-itxi+1+ηi-1W^iTSi+ηixit+zit.According to (31), we obtain (32)χ^˙it=ηiχ~it+zit+W~iTSi.Hence, the virtual control law xi+1∗t is proposed as (33)xi+1∗t=gi-1x-it-Kizit-gi-1Tx-i-1tzi-1t-fix-it+x˙i∗t-ηi-1W^iTSi-χ^it.Choose the Lyapunov-Krasovskii functional candidate as (34)Vi=12ziTtzit+∫t-τitPix-iτdτ+trW~iTΓi-1W~i+12χ~iTtχ~it,where the positive function Pix-iτ is given in the form of (35)Pix-iτ=12hiTx-iτhix-iτ.Similar to Step 1, the time derivative of V˙i can be rewritten as (36)V˙i=-ziTtKizit+ziTtgix-itzi+1t-ziTtgi-1Tx-i-1tzi-1t-ziTtγit+ziTthix-it-τi+Pix-it-Pix-it-τi+σitrW~iTΓi-1W^i-ηiχ~iTtχ~it-χ~iTtW~iTSi+χ~iTtχ˙it.Step n. Define the error variable znt∈Rm: (37)znt=xnt-xn∗tInvoking (1) and (37), differentiating znt with respect to time yields (38)z˙nt=fnx-nt+gnx-ntut+hnx-nt-τ+Δfnx-nt-x˙n∗t.Since Δfnx-nt is unknown, the RBFNN is used to approximate Δfnx-nt. W^n indicates the estimate of Wn∗, and εn∗ is the approximate error. W~n represents the estimated error, which is defined as W~n=Wn∗-W^n. Define the variable χn=εn∗+dnt. (39)Δfnx-nt+γnt=ηn-1Wn∗TSn+εn∗,where γnt=γn,1t,…,γn,mtT, and (40)γn,jt=16znjttanh2⁡znjtvnhnj2x-nt,j=1,…,m.Substituting (39) into (1) yields (41)x˙nt=fnx-nt+gnx-ntut+hnx-nt-τn-γnt+ηn-1Wn∗TSn+χnt.Invoking (38) and (39), one obtains (42)z˙nt=fnx-nt+gnx-ntut+hnx-nt-τn-γnt+ηn-1Wn∗TSn+χnt-x˙n∗t.Construct updating law of the RBFNN (43)W^˙nt=Γnηn-1zntSn-σnΓn-1W^nt,where σn∈R, σn>0, is a design parameter. Furthermore, the DOB can be chosen as (44)χ^nt=wnt+ηnxntw˙nt=-ηnwnt-ηnfnx-nt+gnx-ntut+ηn-1W^nTSn+ηnxnt+znt.According to (44), we obtain (45)χ^˙nt=ηnχ~nt+znt+W~nTSn.Therefore, design the control law ut as (46)ut=gn-1x-nt-Knznt-gn-1Tx-n-1tzn-1t-fnx-nt+x˙n∗t-ηn-1W^nTSn-χ^nt,where Kn∈Rm×m, Kn>0 is design matrix. Choose the Lyapunov-Krasovskii functional candidate as (47)Vn=12znTtznt+∫t-τntPnx-nτdτ+trW~nTΓn-1W~n+12χ~nTtχ~ntand the positive function Pnx-nτ can be designed as follows: (48)Pnx-nτ=12hnTx-nτhnx-nτ.According to the derivatives in Step 1 and Step i, we have (49)V˙n=-znTtKnznt-znTtgn-1Tx-n-1tzn-1t+znTthnx-nt-τn-znTtγnt+Pnx-nt-Pnx-nt-τn-σntrW~nTΓn-1W^n-ηnχ~nTtχ~nt-χ~nTtW~nTSn+χ~nTtχ˙nt.From the above inductive design procedure, moreover, we can conclude the following theorem.


## 3.2. Stability Analysis

Theorem 7.
Considering the system dynamics described by (1), under the adaptive neural updated laws (14), (30), and (43), the disturbance observers (15), (31), and (45), and the prescribed performance-based control laws (17), (33), and (46), we can conclude that the trajectories of the closed-loop system are semiglobally uniformly bounded, while the tracking error z~1t converges to a compact set Qz asymptotically, where C and κ are defined in (50).(50)κ=min⁡λminKi-ϑξi+12,ηi-12ϑξi-δSφi2-πi2,σi2C=∑i=1nσi2λminΓi-1Wi∗2+∑i=1n12πiχ˙it2,where Qz can be made as small as desired by appropriately choosing design parameters. λmin represents the minimum eigenvalue of the matrix. λmin represents the minimum eigenvalue of the matrix.Proof.
For analytical purposes, define the total Lyapunov function(51)VΞ=∑i=1nVi,where definition of Vi can be referred to (19), (34), and (47). According to (23), (36), and (49), we obtain (52)V˙Ξ=-∑i=1nziTtKizit-∑i=1nηiχ~iTtχ~it+∑i=1nziTthix-it-τi+∑i=1nziTtχ~it-∑i=1nziTtγit-∑i=1nχ~iTtW~iTSi+∑i=1nχ~iTtχ˙it+∑i=1nPix-it-Pix-it-τi-∑i=1nσitrW~iTΓi-1W^i.Considering the fact that (53)ziTthix-it-τi≤12ziTtzit+12hiTx-it-τihix-it-τiand Pix-it-τi=1/2hiTx-it-τihix-it-τi, we have (54)V˙Ξ≤-∑i=1nziTtKizit-∑i=1nηiχ~iTtχ~it+∑i=1nziTtχ~it-∑i=1nziTtγit-∑i=1nχ~iTtW~iTSi+∑i=1nχ~iTtχ˙it+12∑i=1nziTtzit+∑i=1nPix-it-∑i=1nσitrW~iTΓi-1W^i.Then we have the following results: (55)V˙Ξ≤-∑i=1nziTtKi-12Im×mzit-∑i=1nηiχ~iTtχ~it+∑i=1nziTtχ~it-∑i=1nziTtγit-∑i=1nχ~iTtW~iTSi+∑i=1nχ~iTtχ˙it+∑i=1nPix-it-∑i=1nσitrW~iTΓi-1W^i.In addition, it is clear that there exists Si≤δSi, δSi>0; we have the following facts: (56)σitrW~iTΓi-1W^i≤-σi2trW~iTΓi-1W~i+σi2trWi∗TΓi-1Wi∗ziTtχ~it≤ϑξi2ziTtzit+12ϑξiχ~it2χ~iTtW~iTSi≤δSiφi2χ~it2+δSi2φiW~i2χ~iTtχ˙it≤πi2χ~it2+12πiχ˙it2,where ϑξi>0 and φi>0 are the constants, and then we have (57)V˙Ξ≤-∑i=1nλminKi-ϑξi+12zit-∑i=1nηi-12ϑξi-δSiφi2-πi2χ~it-∑i=1nσi2trW~iTΓi-1W~i+12∑i=1n∑j=1m1-16tanh2⁡zijtvihijTx-ithijx-it+∑i=1nσi2trWi∗TΓi-1Wi∗+∑i=1n12πiχ˙it.Invoking Lemma 5, we obtain (58)V˙Ξ≤-κVΞ+C.κ>0,C>0 are defined in (50). Therefore, according to Lemma 4, we can conclude that the solution of the closed-loop system remains within a compact subset.


## 4. Application to Unmanned Helicopter System

In this section, we will apply the proposed robust adaptive control strategy to solve the problem of attitude tracking for a class of uncertain unmanned helicopter systems.The unmanned helicopters rigid-body dynamics consist of two parts, attitude angular dynamics and flapping dynamics. The unsteady aerodynamics bring out the time-delay nonlinear uncertainty; thus in this section we consider the attitude control for attitude control subject to time delay. Firstly, the attitude angular and angular velocity dynamics can be described as [26] (59)Θ˙t=RΘtΩtIΩ˙t=-Ωt×IΩt+A-tυct+hΘt,Ωt+B-t+ΔΩΘt,Ωt+dΩt,where Θ=ψ,θ,ϕT∈R3 indicates the attitude angle and Ω=p,q,rT∈R3 represents the attitude angle velocity. dΩt∈R3 stands for the disturbance, and ΔΩ∈R3 stands for the uncertainties; hΘt,Ωt∈R3 means the unknown nonlinear block of time delay. υc=a,b,TTT∈R3, RΘt denotes (60)RΘt=1sin⁡ϕttan⁡θtcos⁡ϕttan⁡θt0cos⁡ϕt-sin⁡ϕt0sin⁡ϕtcos⁡θtcos⁡ϕtcos⁡θt.A- and B- denote (61)A-=-GR-zRTR+KβzT-zRTR+KβGR0yRTRxRTR-xTB-=-yRTRxRTR-GRGR=CRTR1.5+DR.In the unmanned helicopter attitude control loop, the main rotor thrusts TR are fixed. xR,yR,zRT∈R3 indicates the location of the main rotor relative to the center of gravity; xT,yT,zTT∈R3 indicates the location of the tail rotor relative to the center of gravity. Kβ∈R means the stiffness of the rotor hub. CR and DR are constants, which are associated with the antitorque. The control design procedure can be presented as follows.Step 1.
Define the attitude angle tracking errors in the form ofz~ψt=ψt-ψrt, z~θt=θt-θrt, z~ϕt=ϕt-ϕrt, and z~Θt=z~ψt,z~θt,z~ϕtT, where ψrt, θrt, and ϕrt are the reference trajectories and Θr=[ψr(t),θr(t),ϕr(t)]T. Then the time derivative of z~Θt is achieved as (62)z~Θ=RΘtΩt-Θ˙rt.According to the analysis in the previous section, the prescribed performance function is chosen as (63)zΘjt=12ln⁡α_j+z~Θjt/ρjtα-j-z~Θjt/ρjt,j=1,…,3,and the definition of ρjt can be referred to in (6). Choose the virtual control signal as (64)Ω∗t=R-1Θt-MΘ-1tKΘzΘt+Θ˙rt+MΘ-1tNΘt.Choose the Lyapunov function: (65)VΘ=zΘTtzΘt2.Step 2.
Define the tracking errorszpt=pt-pr∗t, zqt=qt-qr∗t, zrt=rt-rr∗t, and Ω∗t=pr∗t,qr∗t,rr∗tT. Using the neural network to compensate the uncertainties and time-delay terms, such that ΔΩΘt,Ωt+γΩt=ηΩ-1WΩ∗TSΩ+εΩ∗, assume that εΩ∗ is the optimal approximate error, and define χΩt=εΩ∗+dΩt. Design the neural updated law as (66)W^˙Ωt=ΓΩzΩtSΩΘt,Ωt-σΩΓΩ-1W^Ωt.The DOB can be constructed in the form of (67)χ^Ωt=wΩt+ηΩΩtw˙Ωt=-ηΩwΩt-ηΩ-Ωt×IΩt+A-tυct+B-t+ηΩ-1W^ΩTSΩ+ηΩΩt+zΩt.Based on the above design, the adaptive control law is achieved as (68)υct=A--1t-KΩzΩt-RTΘzΘt+Ω˙∗t+Ωt×IΩt-B-t-ηΩ-1W^ΩTSΩ-χ^Ωt.Choose the Lyapunov function: (69)VΩ=12ΩTtΩt+∫t-τtPΘtτ,Ωtτdtτ+trW~ΩTΓΩ-1W~Ω+12χ~ΩTtχ~Ωt.Similar to Theorem 7, we can conclude the following theorem.Theorem 8.
For the unmanned helicopter system (59), combining the DOB (67) and neural network update law (66), the control procedure can be achieved as (64) and (68); thus the closed-loop signals are bounded, and the output trajectories are satisfying the prescribed performance conditions.
For Theorem8, the detailed proof can be referred to in Theorem 7 in the previous section.


## 5. Simulation Results

In this section, the numerical simulation about unmanned helicopter system is utilized for illustrating the effectiveness of the previous control method in this paper. Consider the unmanned helicopter system with the following parameters (Table1) [24, 26].Table 1
Helicopter parameters.
Symbol
Unit
Number
I
kg⋅m2
diag
⁡
0.18,0.34,0.28
m
kg
8.2
g
m/sec2
9.81
x
T
m
-
0.91
z
T
m
-
0.08
z
R
m
-
0.235
x
R
m
0
y
R
m
0
z
R
m
0
K
β
N⋅m/rad
52
CR
m
/
N
0.004452
DR
N
⋅
m
0.6304Now let us look at the design of adaptive controller design, the DOB design parameters are chosen asηΩ=6. The neural update law has been given by 70, and its parameters are chosen as ΓΩ=0.1·diag⁡1,1,1,1,1,1,1, σΩ=10, μ=[-3,-2,-1,0,1,2,3], and cΩ=1,1,1,1,1,2,3, and the controller design parameters are chosen as KΘ=5,5,5 and KΩ=10,10,10. The simulation environment is built under Matlab, and the simulation step is chosen as τ=0.02. The initial conditions are chosen as follows: ψ0=1 rad, ψ0=0 rad, and ψ0=0 rad. The time-delay block is chosen in the form of hΘt,Ωt=Ωt-τ2+sin⁡0.1·Ωt-3τ. The external disturbance is chosen as dΩt=2·sin⁡2·t+0.6.From the curves in Figures1–3, it can be seen that the tracking trajectories soon reach the target trajectories, and the tracking errors are bounded in the appointed region. Figures 4–6 show the attitude angular velocities. Figures 7–9 show the control input signals. From the numerical simulation, we can conclude that the proposed control approach is valid for a class of uncertain unmanned helicopter systems with unsteady aerodynamics. Furthermore, it can be illustrated that the closed-loop system output signals are asymptotically tracking the ideal trajectories, and they are restricted in the region of output constraints.Figure 1
Yaw angle.Figure 2
Pitch angle.Figure 3
Roll angle.Figure 4
Yaw angular velocity.Figure 5
Pitch angular velocity.Figure 6
Roll angular velocity.Figure 7
Control input signalυc1t.Figure 8
Control input signalυc2t.Figure 9
Control input signalυc3t.


## 6. Conclusion

In this paper, an adaptive prescribed performance control procedure has been proposed for a class of nonlinear time-delay systems with uncertainties, external disturbances, and output constraints. Additionally, the robust controller has been applied to unmanned helicopter systems with unsteady aerodynamics. At last, the simulation illustrates that the proposed control approach is valid for the uncertain constrained time-delay system.



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*Source: 8410360-2018-02-13.xml*