# Almost Disturbance Decoupling for a Class of Fractional-Order Nonlinear Systems with Zero Dynamics

Authors: Xiaoping Liu; Yajing Zhao; Caiyun Wang; Huanqing Wang; Yucheng Zhou
Journal: Complexity
 (2020)

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## Abstract

The problem of almost disturbance decoupling is addressed for fractional-order nonlinear systems. A new definition for the norm is proposed to describe the effect of disturbances on the output tracking error for fractional-order systems. Based on the Lyapunov stability theory and the backstepping design method, a tracking controller is constructed to make the output tracking error converge to zero without external disturbances and to attenuate the effect of disturbances on the tracking error at zero initial conditions. In order to validate these theoretical results, a numerical example and two practical examples are given.


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## Body

## 1. Introduction

Recently, there has been a continuous growth in the research of fractional-order systems (FOSs). Thanks to the great efforts contributed by researchers, a large number of research results have been published on FOSs, see [1, 2]. It has been proved that fractional-order calculus can model systems more concisely than integer-order calculus [3]. Therefore, FOSs have been applied in numerous areas.The output tracking control problem is one of the most important research topics. It is worth mentioning that the output tracking control has been widely applied in engineering systems including industry and economics [4–6]. As for FOSs, few research outcomes are published. An output tracking controller for a class of fractional-order positive switched systems is developed via an observer-based controller method [7]. In [8], a novel adaptive iterative learning sliding mode controller is proposed to control FOSs. What is more, the output tracking controllers can also be developed with the help of backstepping technique for FOSs. For instance, Wang et al. addressed an adaptive fuzzy output tracking control law for a class of uncertain FOSs with backstepping technique [9]. An output feedback neural network tracking controller for fractional-order nonlinear systems is constructed with the backstepping method [10].Over the past decades, backstepping has become a widely used technique for adaptive nonlinear control. The technique was comprehensively addressed [11], and the basic idea of the backstepping scheme is to design a controller recursively by using a set of virtual controllers. The integer-order backstepping technique has been widely applied in many fields [12, 13]. An adaptive finite-time fuzzy funnel controller is presented for nonaffine nonlinear systems via backstepping technique [14]. An event-triggered adaptive tracking controller based on a new funnel function for uncertain nonlinear systems is proposed by using backstepping technique [15]. Few research outcomes applying backstepping technique to FOSs are published. In [16], synchronization of two fractional-order chaotic Coullet systems is investigated based on the adaptive backstepping. A novel adaptive control technique named adaptive fractional-order backstepping is proposed for fractional-order nonlinear systems (FONSs) with uncertain constant parameters [17]. In [18], Ding et al. extended the classical backstepping to fractional-order backstepping for stabilising FONSs. Then, the pseudo-state stabilization problem of commensurate FONSs is constructed via the fractional-order backstepping scheme [19]. Wei et al. proposed a new technique to construct an adaptive backstepping controller for noncommensurate FOSs with the triangular form [20], and Wei et al. originally presented an adaptive output feedback control law for fractional-order systems by using the backstepping method [21]. Owing to most of the results which construct Lyapunov functions limited to the use of the frequency distribution model or linear matrix inequality (LMI), and the fractional-order backstepping techniques are still to be further studied.It is well known that disturbances exist widely in industrial processes. The issue of disturbance decoupling has drawn great attention. The exact disturbance decoupling problem is to design a feedback controller so that the disturbance does not affect the output of the closed-loop system, which can be solved by the state or output feedback under some strict conditions [22–24]. However, for many practical systems, the exact disturbance decoupling problem is not solvable because the strict conditions are not satisfied. Therefore, the almost disturbance decoupling (ADD) problem is proposed by Weiland and Willems [25], which is to design a feedback controller such that the closed-loop system is stable and the L2 gain from the disturbance to the output is less than or equal to an arbitrary positive constant. Up to now, there have been many research outcomes for the ADD problem for both linear and nonlinear control systems. For example, the ADD problem for SISO nonlinear systems is proposed [26]. The ADD problem of MIMO nonlinear systems and its application to chemical processes are addressed in [27]. In [28], Lin et al. discussed the problem of ADD for nonlinear systems with unstable zero dynamics. Also, in [29], Lin et al. developed a global result about the ADD problem for nonlinear systems with unstable zero dynamics. In [30], the authors investigated the problem of nonlinear output feedback tracking with almost disturbance decoupling. What is more, the ADD problem for high-order nonlinear systems is addressed by [31]. Tracking and almost disturbance decoupling for nonlinear systems with uncertainties are solved by Chen et al. [32]. Chen et al. [33] developed a direct adaptive fuzzy almost disturbance decoupling control method for nonlinear MIMO systems with unknown nonlinearities. For more research results on the ADD problem, see [34–38]. However, the research studies mentioned above are conducted only for integer-order systems. To the best of our knowledge, there are no reports on the ADD problem for FOSs.The active disturbance rejection control (ADRC) problem, which utilizes an estimation and cancellation method to attenuate the negative impact of internal unmodelled dynamics and external disturbances, has been paid extensive attention recently. The ADRC problem has been addressed for nonlinear systems [39–44].Motivated by the aforementioned reviews, this article develops an adaptive backstepping control algorithm to solve the problem of ADD for a class of fractional-order nonlinear systems. To this end, the following two challenges have to be overcome:(1)
Aghababa [45] has proved that Lin et al. [46] mistakenly applied the integer-order integral to deal with the fractional-order differential, which causes that the proposed controller in [46] cannot guarantee the H∞ performance of the closed-loop system, which implies that the L2-norm cannot be directly used to solve the ADD problem. To solve this problem, a new form of norm based on the definition of Caputo fractional integral is proposed.(2)
When the backstepping design method is used, the fractional-order derivative of the virtual controller is required, which is a function of state variables. Therefore, the fractional-order derivative of a composite function is necessary. Unlike the integer-order derivative, the fractional-order derivative of a composite function is so complicated that it cannot be used to the fractional-order derivative of the virtual controller. To avoid such a challenge, the nonlinear functions in all the subsystems but last one are assumed to be linear.Compared with the existing results, the main contributions are listed as follows:(1)
The problem of ADD is first addressed for a class of fractional-order nonlinear systems with external disturbances and zero dynamics.(2)
A new form of the norm based on the definition of Caputo fractional integral is proposed in order to prove theH∞ tracking performance for FOSs.(3)
With the help of backstepping technique, a tracking feedback control law is constructed, which ensures that the effect of external disturbances on the output tracking error is attenuated. A new form of Lyapunov function is chosen without the use of the frequency distribution model or LMI.The remaining parts are arranged as follows. In Section2, several definitions and useful lemmas relevant to the fractional-order calculus are shown. Section 3 proposes the problem formulation and gives the aim of this paper. In Section 4, the main results of this paper are presented. Section 5 gives a numerical example and two practical examples to further support the theoretical results. In the end, the conclusions are summarized in Section 6.


## 2. Preliminaries

### 2.1. Fractional Calculus

Definition 1 (Caputo fractional derivative [3]).
Letft have continuous derivatives up to order n−1, i.e., fn−1t is a real continuously differentiable function. The Caputo fractional-order derivative of order α of ft is defined by(1)Dt0tαft=1Γn−α∫t0tfnσt−σn−α−1dσ,where 0≤n−1<α≤n, n∈ℕ, and Γ⋅ is the gamma function.
For simplicity in notations,Dt0tα is denoted as Dα for t0=0.Definition 2 (Caputo fractional integral [3]).
Assume thatft is an integrable function. Its Caputo fractional integral of order α is defined by(2)Dt0t−αft=1Γα∫t0tfσt−σα−1dσ,where α>0 and t0 is the lower terminal.
For convenience in notations,Dt0t−α is denoted by D−α for t0=0.
Properties of the Caputo fractional calculus, which are used in the paper, are shown as Lemmas1 and 2.Lemma 1 (an inequality of the Caputo fractional derivative [2]).
For a differentiable functionxt∈ℝn, a positive-definite matrix P∈ℝn×n, and 0<α<1, the inequality(3)12DαxTtPxt≤xTtPDαxt,is true for any t≥0.Lemma 2 (see [47]).
Assume that0≤n−1<α≤n and ft is differentiable up to order n−1, that is, fn−1t is a differentiable function. Then,(4)Dt0t−αDt0tαft=ft−∑k=0n−1fkt0k!t−t0k.
In particular, for a continuous functionft and 0<α≤1, the following holds:(5)Dt0t−αt0Dtαft=ft−ft0.


### 2.2. Stability of Fractional-Order Systems

Consider the following FOS:(6)Dt0tαxt=fx,t,where t stands for the time and 0<α≤1.Definition 3.
A functionβ:0,t⟶0,∞ is said to be a class-K function if it is continuous and strictly increasing with β0=0.
Li et al. [48] extended the Lyapunov direct method to FOSs, which is shown as Lemma 3.Lemma 3 (see [48]).
If there exist class-K functions βi i=1,2,3 and a function Vx,t so that(7)β1x≤Vx,t≤β2x,and(8)Dt0tαVx,t≤−β3x,with 0<α<1, then the equilibrium point x=0 of (6) is asymptotically stable.Lemma 4 (Young’s inequality).
The inequality(9)xTy≤εppxp+1qεqyq,holds for any x,y∈ℝr, ε>0, p>1, and q>1 with p−1q−1=1.


## 2.1. Fractional Calculus

Definition 1 (Caputo fractional derivative [3]).
Letft have continuous derivatives up to order n−1, i.e., fn−1t is a real continuously differentiable function. The Caputo fractional-order derivative of order α of ft is defined by(1)Dt0tαft=1Γn−α∫t0tfnσt−σn−α−1dσ,where 0≤n−1<α≤n, n∈ℕ, and Γ⋅ is the gamma function.
For simplicity in notations,Dt0tα is denoted as Dα for t0=0.Definition 2 (Caputo fractional integral [3]).
Assume thatft is an integrable function. Its Caputo fractional integral of order α is defined by(2)Dt0t−αft=1Γα∫t0tfσt−σα−1dσ,where α>0 and t0 is the lower terminal.
For convenience in notations,Dt0t−α is denoted by D−α for t0=0.
Properties of the Caputo fractional calculus, which are used in the paper, are shown as Lemmas1 and 2.Lemma 1 (an inequality of the Caputo fractional derivative [2]).
For a differentiable functionxt∈ℝn, a positive-definite matrix P∈ℝn×n, and 0<α<1, the inequality(3)12DαxTtPxt≤xTtPDαxt,is true for any t≥0.Lemma 2 (see [47]).
Assume that0≤n−1<α≤n and ft is differentiable up to order n−1, that is, fn−1t is a differentiable function. Then,(4)Dt0t−αDt0tαft=ft−∑k=0n−1fkt0k!t−t0k.
In particular, for a continuous functionft and 0<α≤1, the following holds:(5)Dt0t−αt0Dtαft=ft−ft0.


## 2.2. Stability of Fractional-Order Systems

Consider the following FOS:(6)Dt0tαxt=fx,t,where t stands for the time and 0<α≤1.Definition 3.
A functionβ:0,t⟶0,∞ is said to be a class-K function if it is continuous and strictly increasing with β0=0.
Li et al. [48] extended the Lyapunov direct method to FOSs, which is shown as Lemma 3.Lemma 3 (see [48]).
If there exist class-K functions βi i=1,2,3 and a function Vx,t so that(7)β1x≤Vx,t≤β2x,and(8)Dt0tαVx,t≤−β3x,with 0<α<1, then the equilibrium point x=0 of (6) is asymptotically stable.Lemma 4 (Young’s inequality).
The inequality(9)xTy≤εppxp+1qεqyq,holds for any x,y∈ℝr, ε>0, p>1, and q>1 with p−1q−1=1.


## 3. Problem Formulation

Consider a fractional-order strict-feedback nonlinear system with unknown disturbances and zero dynamics defined by(10)Dαzt=f0zt,Dαx1t=x2t+f1x¯1t+P1ωt,Dαx2t=x3t+f2x¯2t+P2ωt,⋮Dαxn−1t=xnt+fn−1x¯n−1t+Pn−1ωt,Dαxnt=fxt+gxtut+fnx¯nt+Pnωt,yt=x1t,where 0<α<1, zt denotes the zero dynamics of (10), and f0zt is a known nonlinear function with respect to zt. xt=x1t,…,xntT∈ℝn and x¯it=x1t,…,xitT∈ℝi with xit being the state of the system, while yt∈ℝ and ut∈ℝ are the system output and input, respectively. ωt∈ℝr represents the disturbance signal. fix¯it=aix¯it∈ℝi=1,…,n with ai being a known vector. f⋅ and g⋅ are assumed to be nonlinear and known functions. Pi∈ℝ1×r is a known vector for i=1,…,n. For convenience, denote fxt, gxt, f0zt, fix¯it, x¯it, xit, and zt as f, g, f0, fi, x¯i, xi, and z, respectively.The aim is to construct a state feedback controller to drive the outputyt to track a desired reference signal yrt, i.e., limt⟶∞yt−yrt=0 when ωt=0 and to have an H∞ tracking performance for the zero initial state, i.e.,(11)yt−yrtα≤γωtα,where ⋅α is defined later and γ>0.


## 4. Main Results

### 4.1. TheLα Norm

In this section, a new form of norm namedLα norm based on the definition of Caputo fractional integral is presented.Definition 4.
TheLα norm of a function ωt is defined by(12)ωtα=limt⟶∞∫0tωTσt−σα−1ωσdσ,where 0<α<1.
ThisLα norm satisfies three conditions of the norm, i.e., nonnegative (equation (13)), homogeneity (equation (14)), and triangle inequality (equation (15)).Lemma 5.
TheLα norm satisfies the following properties:(i)
Nonnegative:(13)ωtα≥0;ωtα=0,ifandonlyifωt=0.(ii)
Homogeneity:(14)kωtα=kωtα.(iii)
Triangle inequality:(15)ω1t+ω2tα≤ω1tα+ω2tα.Proof.
(i)
According to Definition (12), it can be easily verified that (13) is true.(ii)
Clearly,(16)kωtα=limt⟶∞∫0tk2ωTσt−σα−1ωσdσ=kωtα.(iii)
For convenience, denoteωt as ω. It can be seen that the proof of condition (15) is equivalent to verify(17)ω1+ω2α2≤ω1α+ω2α2.
On the one hand,(18)ω1+ω2α2=limt⟶∞∫0tω1+ω2Tt−σα−1ω1+ω2dσ=limt⟶∞∫0tω1Tω1+2ω1Tω2+ω2Tω2t−σα−1dσ=ω1α2+ω2α2+2limt⟶∞∫0tω1Tt−σα−1ω2dσ.
On the other hand,(19)ω1α+ω2α2=ω1α2+2ω1αω2α+ω2α2=ω1α2+ω2α2+2limt⟶∞∫0tω1Tt−σα−1ω1dσ⋅∫0tω2Tt−σα−1ω1dσ.
It is a simple exercise to show that the proof of (17) is equivalent to prove(20)∫0tω1Tt−σα−1ω2dσ2≤∫0tω1Tt−σα−1ω1dσ⋅∫0tω2Tt−σα−1ω2dσ.
In order to verify (20), we construct a quadratic function of u as follows:(21)Φu=u2∫0tω1Tσt−σα−1ω1σdσ+2u∫0tω1Tσt−σα−1ω2σdσ+∫0tω2Tσt−σα−1ω2σdσ.
After a little manipulation, one can get the following result:(22)Φu=∫0tu2ω1Tσt−σα−1ω1σ+2uω1Tσt−σα−1ω2σ+ω2Tσt−σα−1ω2σdσ=∫0tt−σα−1/2uω1σ+ω2σT⋅uω1σ+ω2σdσ≥0.
It is easily seen from (21) and (22) that the quadratic function intersects the u−axis at most one point, which implies that(23)2∫0tω1Tσt−σα−1ω2σdσ2−4∫0tω1Tσt−σα−1ω1σdσ⋅∫0tω2Tσt−σα−1ω2σdσ≤0.which is equivalent to (20).


### 4.2. Control Laws forH∞ Tracking Performance

Assumption 1.
For the zero dynamicsDαz=f0z in system (10), there exists a positive-definite differentiable function V0 so that its Caputo fractional derivative with order 0<α<1 satisfies the following inequality:(24)DαV0<0.Assumption 2.
The known nonlinear functiong⋅ satisfies the following inequality:(25)g⋅>0.Assumption 3.
The desired reference signalyrt and D˜i are bounded and piecewise continuous, where D˜i=DαDαDα⋯Dα︸iyr,i=0,1,2,…,n.
Define the control law by(26)u=1g−f−knzn−zn−1−fn+∑s=1n−1∂αn−1∂xsxs+1+fs+∑s=0n−1∂αn−1∂D^sD^s+1−P^nP^nT4γnzn,and the virtual control laws by(27)α0=yrt,α1=−k1z1−f1+Dαα0−P1P1T4γ1z1,…αi=−kizi−zi−1−fi+∑s=1i−1∂αi−1∂xsxs+1+fs+∑s=0i−1∂αi−1∂D^sD^s+1−P^iP^iT4γizi,i=2,…,n−1,where(28)z1=x1−α0,zi=xi−αi−1,i=2,3,…,n,(29)P^i=Pi−∑s=1i−1∂αi−1∂xsPs,i=2,3,…,n,and(30)D^i=DαDαDα…Dαα0︸i,i=0,1,2,…,n.Theorem 1.
With Assumptions1–3, virtual control laws (26), and feedback controller (25), the closed-loop system corresponding to (10) has the following properties:(1)
yt asymptotically tracks the desired reference signal yrt, i.e.,(31)limt⟶∞yt−yrt=0,whenωt=0.(2)
yt achieves the H∞ tracking performance, i.e.,(32)yt−yrtα≤γωtα,for the zero initial condition whenωt≠0, where γ>0.Proof.
The proof is given by constructing virtual control laws and a control law based on backstepping design approach.Step 1: select the following Lyapunov function:(33)V1=12z12.Then, the fractional derivative ofV1 is given by(34)DαV1≤z1Dαz1=z1Dαx1−Dαα0=z1x2+f1+P1ω−Dαα0=z1z2+α1+f1+P1ω−Dαα0=z1z2+z1α1+f1−Dαα0+z1P1ω,where the first inequality is generated from (3).According to Lemma4, let p=q=2 and ε=1/2γ; then, (34) can be estimated by(35)DαV1≤z1z2+γ1ωTω+z1α1+f1−Dαα0+z1P1P1T4γ1z1=z1z2+γ1ωTω+z1α1+f1−Dαα0+P1P1T4γ1z1.Then, choose a virtual control law as(36)α1=−k1z1−f1+Dαα0−P1P1T4γ1z1,wherek1 and γ1 are positive constants and can be designed by us.Furthermore, substituting (36) into (35) results in(37)DαV1≤−k1z12+z1z2+γ1ωTω.Step 2:Define a Lyapunov function as(38)V2=V1+12z22.It follows from (28) and (27) that(39)Dαz2=Dαx2−Dαα1=x3+f2+P2ω−Dαα1,with(40)Dαα1=∂α1∂x1x2+f1+P1ω+∂α1∂α0Dαα0+∂α1∂Dαα0DαDαα0.Substituting (40) into (39) yields(41)z2Dαz2=z2x3+f2+P^2ω−∂α1∂x1x2+f1−∑s=01∂α1∂D^sD^s+1.Similar to (35), applying Lemma 4 to P^2ω, (41) can be estimated by(42)z2Dαz2≤z2x3+f2−∂α1∂x1x2+f1−∑s=01∂α1∂D^sD^s+1+P^2P^2T4γ2z2+γ2ωTω.Then, the fractional derivative ofV2 is given by(43)DαV2≤DαV1+z2Dαz2≤−k1z12+∑s=12γsωTω+z2z3+z2α2+z1+f2+P^2P^2T4γ2z2−∂α1∂x1x2+f1−∑s=01∂α1∂D^sD^s+1,where the first inequality is based on (37) and (3), respectively.Similar to Step 1, the virtual control law is chosen as(44)α2=−k2z2−z1−f2+∂α1∂x1x2+f1+∑s=01∂α1∂D^sD^s+1−P^2P^2T4γ2z2,wherek2 is a positive constant. With virtual controller (43), (44) can be expressed as(45)DαV2≤−k1z12−k2z22+z2z3+∑s=12γsωTω.Stepi (i = 3, …, n − 1):In this step, we repeat the procedure in a recursive way. It follows from (27) and (28) that(46)Dαzi=Dαxi−Dααi−1=xi+1+fi+Piω−Dααi−1,with(47)Dααi−1=∑s=1i−1∂αi−1∂xsxs+1+fs+Psω+∑s=0i−1∂αi−1∂D^sD^s+1.Substituting (47) into (46) yields(48)ziDαzi=zixi+1+fi+P^iω−∑s=1i−1∂αi−1∂xsxs+1+fs−∑s=0i−1∂αi−1∂D^sD^s+1≤zixi+1+fi−∑s=1i−1∂αi−1∂xsxs+1+fs−∑s=0i−1∂αi−1∂D^sD^s+1+P^iP^iT4γizi+γiωTω,where the inequality is based on Lemma4 with p=q=2 and ε=1/2γ.With (48), it can be derived that the Lyapunov function(49)Vi=Vi−1+12zi2has the following fractional derivative:(50)DαVi≤DαVi−1+ziDαzi≤−∑s=1i−1kszs2+zizi+1+∑s=1iγsωTω+ziαi+zi−1+fi−∑s=1i−1∂αi−1∂xsxs+1+fs−∑s=0i−1∂αi−1∂D^sD^s+1+P^iP^iT4γizi.The virtual control law is chosen as(51)αi=−kizi−zi−1−fi+∑s=1i−1∂αi−1∂xsxs+1+fs+∑s=0i−1∂αi−1∂D^sD^s+1−P^iP^iT4γizi,whereki is a positive constant. Next, substituting (51) into (50) results in(52)DαVi≤−∑s=1ikszs2+zizi+1+∑s=1iγsωTω.Stepn:In the final step, select a Lyapunov function as(53)Vn=Vn−1+12zn2.It follows from (27) and (28) that(54)Dαzn=Dαxn−Dααn−1=f+gu+fn+Pnω−Dααn−1,with(55)Dααn−1=∑s=1n−1∂αn−1∂xsxs+1+fs+Psω+∑s=0n−1∂αn−1∂D^sD^s+1.Substituting (55) into (54) yields(56)znDαzn=znf+gu+fn+P^nω−∑s=1n−1∂αn−1∂xsxs+1+fs−∑s=0n−1∂αn−1∂D^sD^s+1≤znf+gu+fn−∑s=1n−1∂αn−1∂xsxs+1+fs−∑s=0n−1∂αn−1∂D^sD^s+1+P^nP^nT4γnzn+γnωTω,where the inequality is based on Lemma4 with p=q=2 and ε=1/2γ.Then, the fractional-order derivative ofVn can be estimated by(57)DαVn≤DαVn−1+znDαzn≤−∑s=1n−1kszs2+∑s=1nγsωTω+znf+gu+fn+zn−1+P^nP^nT4γnzn−∑s=1n−1∂αn−1∂xsxs+1+fs−∑s=0n−1∂αn−1∂D^sD^s+1.Introduce the control law as follows:(58)u=1g−f−kn+P^nP^nT4γnzn−zn−1−fn+∑s=1n−1∂αn−1∂xsxs+1+fs+∑s=0n−1∂αn−1∂D^sD^s+1,whereki is a positive constant. Next, substituting (58) into (57) results in(59)DαVn≤−∑s=1nkszs2+∑s=1nγsωTω.(1)
Whenωt=0, DαVn≤0 can be guaranteed. Then, the output signal yt asymptotically tracks the desired reference signal yrt.(2)
Whenωt≠0, (59) can be estimated by(60)DαVn≤−k1z12+γ¯ωTω,whereγ¯=∑s=1nγs.To simplify the notation,Vn is abbreviated to be V. It follows from (53) that V≥0. For the H∞ performance, assume the initial condition is zero. Then, V0=0 is true. Calculate the Caputo fractional-order integral with order α of (60):(61)V=D−αDαV≤D−α−k1z1Tz1+γ¯ωTω.It means that(62)0≤V≤D−α−k1z1Tz1+γ¯ωTω,i.e.,(63)D−αk1z1Tz1≤D−αγ¯ωTω.Then,(64)k1Γα∫0tz1Tσt−σα−1z1σdσ≤γ¯Γα∫0tωTσt−σα−1ωσdσ.Lett⟶∞:(65)z1tα≤γ¯k1ωtα,i.e.,(66)yt−yrtα≤γωtα,withγ=γ¯/k1.
It means that the effect of disturbances on the output tracking error is attenuated to a given degreeγ. Therefore, the H∞ tracking performance can be achieved.


## 4.1. TheLα Norm

In this section, a new form of norm namedLα norm based on the definition of Caputo fractional integral is presented.Definition 4.
TheLα norm of a function ωt is defined by(12)ωtα=limt⟶∞∫0tωTσt−σα−1ωσdσ,where 0<α<1.
ThisLα norm satisfies three conditions of the norm, i.e., nonnegative (equation (13)), homogeneity (equation (14)), and triangle inequality (equation (15)).Lemma 5.
TheLα norm satisfies the following properties:(i)
Nonnegative:(13)ωtα≥0;ωtα=0,ifandonlyifωt=0.(ii)
Homogeneity:(14)kωtα=kωtα.(iii)
Triangle inequality:(15)ω1t+ω2tα≤ω1tα+ω2tα.Proof.
(i)
According to Definition (12), it can be easily verified that (13) is true.(ii)
Clearly,(16)kωtα=limt⟶∞∫0tk2ωTσt−σα−1ωσdσ=kωtα.(iii)
For convenience, denoteωt as ω. It can be seen that the proof of condition (15) is equivalent to verify(17)ω1+ω2α2≤ω1α+ω2α2.
On the one hand,(18)ω1+ω2α2=limt⟶∞∫0tω1+ω2Tt−σα−1ω1+ω2dσ=limt⟶∞∫0tω1Tω1+2ω1Tω2+ω2Tω2t−σα−1dσ=ω1α2+ω2α2+2limt⟶∞∫0tω1Tt−σα−1ω2dσ.
On the other hand,(19)ω1α+ω2α2=ω1α2+2ω1αω2α+ω2α2=ω1α2+ω2α2+2limt⟶∞∫0tω1Tt−σα−1ω1dσ⋅∫0tω2Tt−σα−1ω1dσ.
It is a simple exercise to show that the proof of (17) is equivalent to prove(20)∫0tω1Tt−σα−1ω2dσ2≤∫0tω1Tt−σα−1ω1dσ⋅∫0tω2Tt−σα−1ω2dσ.
In order to verify (20), we construct a quadratic function of u as follows:(21)Φu=u2∫0tω1Tσt−σα−1ω1σdσ+2u∫0tω1Tσt−σα−1ω2σdσ+∫0tω2Tσt−σα−1ω2σdσ.
After a little manipulation, one can get the following result:(22)Φu=∫0tu2ω1Tσt−σα−1ω1σ+2uω1Tσt−σα−1ω2σ+ω2Tσt−σα−1ω2σdσ=∫0tt−σα−1/2uω1σ+ω2σT⋅uω1σ+ω2σdσ≥0.
It is easily seen from (21) and (22) that the quadratic function intersects the u−axis at most one point, which implies that(23)2∫0tω1Tσt−σα−1ω2σdσ2−4∫0tω1Tσt−σα−1ω1σdσ⋅∫0tω2Tσt−σα−1ω2σdσ≤0.which is equivalent to (20).


## 4.2. Control Laws forH∞ Tracking Performance

Assumption 1.
For the zero dynamicsDαz=f0z in system (10), there exists a positive-definite differentiable function V0 so that its Caputo fractional derivative with order 0<α<1 satisfies the following inequality:(24)DαV0<0.Assumption 2.
The known nonlinear functiong⋅ satisfies the following inequality:(25)g⋅>0.Assumption 3.
The desired reference signalyrt and D˜i are bounded and piecewise continuous, where D˜i=DαDαDα⋯Dα︸iyr,i=0,1,2,…,n.
Define the control law by(26)u=1g−f−knzn−zn−1−fn+∑s=1n−1∂αn−1∂xsxs+1+fs+∑s=0n−1∂αn−1∂D^sD^s+1−P^nP^nT4γnzn,and the virtual control laws by(27)α0=yrt,α1=−k1z1−f1+Dαα0−P1P1T4γ1z1,…αi=−kizi−zi−1−fi+∑s=1i−1∂αi−1∂xsxs+1+fs+∑s=0i−1∂αi−1∂D^sD^s+1−P^iP^iT4γizi,i=2,…,n−1,where(28)z1=x1−α0,zi=xi−αi−1,i=2,3,…,n,(29)P^i=Pi−∑s=1i−1∂αi−1∂xsPs,i=2,3,…,n,and(30)D^i=DαDαDα…Dαα0︸i,i=0,1,2,…,n.Theorem 1.
With Assumptions1–3, virtual control laws (26), and feedback controller (25), the closed-loop system corresponding to (10) has the following properties:(1)
yt asymptotically tracks the desired reference signal yrt, i.e.,(31)limt⟶∞yt−yrt=0,whenωt=0.(2)
yt achieves the H∞ tracking performance, i.e.,(32)yt−yrtα≤γωtα,for the zero initial condition whenωt≠0, where γ>0.Proof.
The proof is given by constructing virtual control laws and a control law based on backstepping design approach.Step 1: select the following Lyapunov function:(33)V1=12z12.Then, the fractional derivative ofV1 is given by(34)DαV1≤z1Dαz1=z1Dαx1−Dαα0=z1x2+f1+P1ω−Dαα0=z1z2+α1+f1+P1ω−Dαα0=z1z2+z1α1+f1−Dαα0+z1P1ω,where the first inequality is generated from (3).According to Lemma4, let p=q=2 and ε=1/2γ; then, (34) can be estimated by(35)DαV1≤z1z2+γ1ωTω+z1α1+f1−Dαα0+z1P1P1T4γ1z1=z1z2+γ1ωTω+z1α1+f1−Dαα0+P1P1T4γ1z1.Then, choose a virtual control law as(36)α1=−k1z1−f1+Dαα0−P1P1T4γ1z1,wherek1 and γ1 are positive constants and can be designed by us.Furthermore, substituting (36) into (35) results in(37)DαV1≤−k1z12+z1z2+γ1ωTω.Step 2:Define a Lyapunov function as(38)V2=V1+12z22.It follows from (28) and (27) that(39)Dαz2=Dαx2−Dαα1=x3+f2+P2ω−Dαα1,with(40)Dαα1=∂α1∂x1x2+f1+P1ω+∂α1∂α0Dαα0+∂α1∂Dαα0DαDαα0.Substituting (40) into (39) yields(41)z2Dαz2=z2x3+f2+P^2ω−∂α1∂x1x2+f1−∑s=01∂α1∂D^sD^s+1.Similar to (35), applying Lemma 4 to P^2ω, (41) can be estimated by(42)z2Dαz2≤z2x3+f2−∂α1∂x1x2+f1−∑s=01∂α1∂D^sD^s+1+P^2P^2T4γ2z2+γ2ωTω.Then, the fractional derivative ofV2 is given by(43)DαV2≤DαV1+z2Dαz2≤−k1z12+∑s=12γsωTω+z2z3+z2α2+z1+f2+P^2P^2T4γ2z2−∂α1∂x1x2+f1−∑s=01∂α1∂D^sD^s+1,where the first inequality is based on (37) and (3), respectively.Similar to Step 1, the virtual control law is chosen as(44)α2=−k2z2−z1−f2+∂α1∂x1x2+f1+∑s=01∂α1∂D^sD^s+1−P^2P^2T4γ2z2,wherek2 is a positive constant. With virtual controller (43), (44) can be expressed as(45)DαV2≤−k1z12−k2z22+z2z3+∑s=12γsωTω.Stepi (i = 3, …, n − 1):In this step, we repeat the procedure in a recursive way. It follows from (27) and (28) that(46)Dαzi=Dαxi−Dααi−1=xi+1+fi+Piω−Dααi−1,with(47)Dααi−1=∑s=1i−1∂αi−1∂xsxs+1+fs+Psω+∑s=0i−1∂αi−1∂D^sD^s+1.Substituting (47) into (46) yields(48)ziDαzi=zixi+1+fi+P^iω−∑s=1i−1∂αi−1∂xsxs+1+fs−∑s=0i−1∂αi−1∂D^sD^s+1≤zixi+1+fi−∑s=1i−1∂αi−1∂xsxs+1+fs−∑s=0i−1∂αi−1∂D^sD^s+1+P^iP^iT4γizi+γiωTω,where the inequality is based on Lemma4 with p=q=2 and ε=1/2γ.With (48), it can be derived that the Lyapunov function(49)Vi=Vi−1+12zi2has the following fractional derivative:(50)DαVi≤DαVi−1+ziDαzi≤−∑s=1i−1kszs2+zizi+1+∑s=1iγsωTω+ziαi+zi−1+fi−∑s=1i−1∂αi−1∂xsxs+1+fs−∑s=0i−1∂αi−1∂D^sD^s+1+P^iP^iT4γizi.The virtual control law is chosen as(51)αi=−kizi−zi−1−fi+∑s=1i−1∂αi−1∂xsxs+1+fs+∑s=0i−1∂αi−1∂D^sD^s+1−P^iP^iT4γizi,whereki is a positive constant. Next, substituting (51) into (50) results in(52)DαVi≤−∑s=1ikszs2+zizi+1+∑s=1iγsωTω.Stepn:In the final step, select a Lyapunov function as(53)Vn=Vn−1+12zn2.It follows from (27) and (28) that(54)Dαzn=Dαxn−Dααn−1=f+gu+fn+Pnω−Dααn−1,with(55)Dααn−1=∑s=1n−1∂αn−1∂xsxs+1+fs+Psω+∑s=0n−1∂αn−1∂D^sD^s+1.Substituting (55) into (54) yields(56)znDαzn=znf+gu+fn+P^nω−∑s=1n−1∂αn−1∂xsxs+1+fs−∑s=0n−1∂αn−1∂D^sD^s+1≤znf+gu+fn−∑s=1n−1∂αn−1∂xsxs+1+fs−∑s=0n−1∂αn−1∂D^sD^s+1+P^nP^nT4γnzn+γnωTω,where the inequality is based on Lemma4 with p=q=2 and ε=1/2γ.Then, the fractional-order derivative ofVn can be estimated by(57)DαVn≤DαVn−1+znDαzn≤−∑s=1n−1kszs2+∑s=1nγsωTω+znf+gu+fn+zn−1+P^nP^nT4γnzn−∑s=1n−1∂αn−1∂xsxs+1+fs−∑s=0n−1∂αn−1∂D^sD^s+1.Introduce the control law as follows:(58)u=1g−f−kn+P^nP^nT4γnzn−zn−1−fn+∑s=1n−1∂αn−1∂xsxs+1+fs+∑s=0n−1∂αn−1∂D^sD^s+1,whereki is a positive constant. Next, substituting (58) into (57) results in(59)DαVn≤−∑s=1nkszs2+∑s=1nγsωTω.(1)
Whenωt=0, DαVn≤0 can be guaranteed. Then, the output signal yt asymptotically tracks the desired reference signal yrt.(2)
Whenωt≠0, (59) can be estimated by(60)DαVn≤−k1z12+γ¯ωTω,whereγ¯=∑s=1nγs.To simplify the notation,Vn is abbreviated to be V. It follows from (53) that V≥0. For the H∞ performance, assume the initial condition is zero. Then, V0=0 is true. Calculate the Caputo fractional-order integral with order α of (60):(61)V=D−αDαV≤D−α−k1z1Tz1+γ¯ωTω.It means that(62)0≤V≤D−α−k1z1Tz1+γ¯ωTω,i.e.,(63)D−αk1z1Tz1≤D−αγ¯ωTω.Then,(64)k1Γα∫0tz1Tσt−σα−1z1σdσ≤γ¯Γα∫0tωTσt−σα−1ωσdσ.Lett⟶∞:(65)z1tα≤γ¯k1ωtα,i.e.,(66)yt−yrtα≤γωtα,withγ=γ¯/k1.
It means that the effect of disturbances on the output tracking error is attenuated to a given degreeγ. Therefore, the H∞ tracking performance can be achieved.


## 5. Simulation Results

In order to verify the excellent properties of the proposed tracking controller, a numerical example is presented as Example 1, and two practical examples are given as Examples 2 and 3.


### 5.1. Example 1

Consider system (10) with α=0.93, f0z=−z3, f1x¯1=0.1x1t, f2x¯2=0.5x1t+x2t, f3x¯3=0.2x1t−0.01x2t+x3t, gx=x12t, fx=x22t, P1=P2=P3=0.1, and ωt=coste−0.5t. The desired reference signal is set to yrt=0.5sinπt/5. The initial state x10,x20,x30T=0,0.3226,−0.2443T and z0=0.1 are considered.The virtual laws are chosen as(67)α1=−k1+P1P1T4γ1z1−f1+Dαα0,α2=−k2+P^2P^2T4γ2z2−z1−f2+∂α1∂x1x2+f1+∑s=01∂α1∂D^sD^s+1.The adaptive control law is determined by(68)u=1g−f−k3+P^3P^3T4γ3z3−z2−f3+∑s=12∂α2∂xsxs+1+fs+∑s=02∂αn−1∂D^sD^s+1,where the design parameters are set to k1=12, k2=10, k3=10, γ1=0.01, γ2=0.2, and γ3=4.Simulation results of Example 1 are shown as follows. Figure1 gives the tracking performance, from which it can be concluded that the asymptotic tracking can be realized. It is obviously seen that the state variables are influenced by external disturbances in Figure 2. Figure 3 indicates the trajectories of the control input.Figure 1
The tracking performance in Example 1.Figure 2
The state responses in Example 1.Figure 3
The control input in Example 1.


### 5.2. Example 2

Consider the fractional-order model of the magnetic leviation (FO-MAGLEV) suspension system in [49]:(69)Dαx1t=x2t+P1ωt,Dαx2t=x3t+P2ωt,Dαx3t=−6233.8x1t−65.2x2t−4.8x3t+0.9212u+P3ωt,yt=x1t,with α=0.93, P1=0.1, P2=0, P3=−65.17∗0.1, yrt=0.5sinπt/5, x10,x20,x30T=0,0.3226,0.046T, and ωt=coste−0.5t.The virtual laws and the adaptive control law are designed as (27) and (26) with the parameters chosen as k1=16, k2=12, k3=10, γ1=0.01, γ2=0.1, and γ3=5. Figures 4–6 demonstrate the simulation results for Example 2. The tracking performance is illustrated in Figure 4. The state responses of Example 2 are shown in Figure 5. Figure 6 presents the control input of Example 2. It is seen that the tracking controller succeeds to make the output tracking error converge to zero without external disturbances and to attenuate the effect of disturbances on the tracking error.Figure 4
The tracking performance in Example 2.Figure 5
The state responses in Example 2.Figure 6
The control input in Example 2.


### 5.3. Example 3

Consider the fractional-order model of the horizontal platform system (FO-HPS) given in [50]:(70)Dαx1t=x2t+P1ωt,Dαx2t=−a¯x2t−bsinx1t+lsinx1tcosx1t+hcosτt+ut+P2ωt,yt=x1t,with α=0.93, P1=0, P2=1, yrt=0.5sinπt/5, a¯=4/3, b=3.776, l=4.6∗10−6, h=3.4/4, τ=1.8, x10,x20T=0,0.3226T, and ωt=coste−0.5t.The virtual laws and the adaptive control law are determined by (27) and (26), where the design parameters are chosen as k1=10, k2=10, γ1=0.01, and γ2=0.05.The simulation results of Example 3 are presented in Figures7–9. The tracking performance of Example 3 is shown in Figure 7. The state responses of Example 3 are given in Figure 8. Figure 9 presents the control input of Example 3. It is observed that the reference signal can be tracked closely by the output signal and the impact of the external disturbances on the output tracking error can be reduced to a certain extent.Figure 7
The tracking performance in Example 3.Figure 8
The state response in Example 3.Figure 9
The control input in Example 3.


## 5.1. Example 1

Consider system (10) with α=0.93, f0z=−z3, f1x¯1=0.1x1t, f2x¯2=0.5x1t+x2t, f3x¯3=0.2x1t−0.01x2t+x3t, gx=x12t, fx=x22t, P1=P2=P3=0.1, and ωt=coste−0.5t. The desired reference signal is set to yrt=0.5sinπt/5. The initial state x10,x20,x30T=0,0.3226,−0.2443T and z0=0.1 are considered.The virtual laws are chosen as(67)α1=−k1+P1P1T4γ1z1−f1+Dαα0,α2=−k2+P^2P^2T4γ2z2−z1−f2+∂α1∂x1x2+f1+∑s=01∂α1∂D^sD^s+1.The adaptive control law is determined by(68)u=1g−f−k3+P^3P^3T4γ3z3−z2−f3+∑s=12∂α2∂xsxs+1+fs+∑s=02∂αn−1∂D^sD^s+1,where the design parameters are set to k1=12, k2=10, k3=10, γ1=0.01, γ2=0.2, and γ3=4.Simulation results of Example 1 are shown as follows. Figure1 gives the tracking performance, from which it can be concluded that the asymptotic tracking can be realized. It is obviously seen that the state variables are influenced by external disturbances in Figure 2. Figure 3 indicates the trajectories of the control input.Figure 1
The tracking performance in Example 1.Figure 2
The state responses in Example 1.Figure 3
The control input in Example 1.


## 5.2. Example 2

Consider the fractional-order model of the magnetic leviation (FO-MAGLEV) suspension system in [49]:(69)Dαx1t=x2t+P1ωt,Dαx2t=x3t+P2ωt,Dαx3t=−6233.8x1t−65.2x2t−4.8x3t+0.9212u+P3ωt,yt=x1t,with α=0.93, P1=0.1, P2=0, P3=−65.17∗0.1, yrt=0.5sinπt/5, x10,x20,x30T=0,0.3226,0.046T, and ωt=coste−0.5t.The virtual laws and the adaptive control law are designed as (27) and (26) with the parameters chosen as k1=16, k2=12, k3=10, γ1=0.01, γ2=0.1, and γ3=5. Figures 4–6 demonstrate the simulation results for Example 2. The tracking performance is illustrated in Figure 4. The state responses of Example 2 are shown in Figure 5. Figure 6 presents the control input of Example 2. It is seen that the tracking controller succeeds to make the output tracking error converge to zero without external disturbances and to attenuate the effect of disturbances on the tracking error.Figure 4
The tracking performance in Example 2.Figure 5
The state responses in Example 2.Figure 6
The control input in Example 2.


## 5.3. Example 3

Consider the fractional-order model of the horizontal platform system (FO-HPS) given in [50]:(70)Dαx1t=x2t+P1ωt,Dαx2t=−a¯x2t−bsinx1t+lsinx1tcosx1t+hcosτt+ut+P2ωt,yt=x1t,with α=0.93, P1=0, P2=1, yrt=0.5sinπt/5, a¯=4/3, b=3.776, l=4.6∗10−6, h=3.4/4, τ=1.8, x10,x20T=0,0.3226T, and ωt=coste−0.5t.The virtual laws and the adaptive control law are determined by (27) and (26), where the design parameters are chosen as k1=10, k2=10, γ1=0.01, and γ2=0.05.The simulation results of Example 3 are presented in Figures7–9. The tracking performance of Example 3 is shown in Figure 7. The state responses of Example 3 are given in Figure 8. Figure 9 presents the control input of Example 3. It is observed that the reference signal can be tracked closely by the output signal and the impact of the external disturbances on the output tracking error can be reduced to a certain extent.Figure 7
The tracking performance in Example 3.Figure 8
The state response in Example 3.Figure 9
The control input in Example 3.


## 6. Conclusion

The ADD problem has been addressed for a type of nonlinear FOSs with time-varying disturbances and zero dynamics. With the help of the backstepping method, a stabilising state feedback control law has been constructed, which guarantees that asymptotic tracking is achieved without external disturbances and the effect of the external disturbances on the output tracking error can be reduced to a certain extent. In addition, a numerical example and two practical examples are given to validate the theoretical results.



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