# An Adaptive Control of Fractional-Order Nonlinear Uncertain Systems with Input Saturation

Authors: Changhui Wang; Mei Liang; Yongsheng Chai
Journal: Complexity
 (2019)

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## Abstract

This paper develops a fractional-order adaptive fuzzy backstepping control scheme for incommensurate fractional-order nonlinear uncertain systems with external disturbances and input saturation. Based on backstepping algorithm, the fuzzy logic system is used to approximate the unknown nonlinear uncertainties in each step of the backstepping, and the fractional-order parameters update laws for fuzzy logic system, unknown parameters, and the external disturbances are proposed. With the aids of the frequency distributed model of fractional integrator for the fractional-order systems in the procedure of controller design, the stability of the closed-loop system is established. To verify the effectiveness and robustness of the proposed controller, two simulation examples are demonstrated at last.


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## Body

## 1. Introduction

In the latest two decades, fractional-order calculus as a generalization of derivation and integration of arbitrary order has attracted more and more attentions and interests in both theory and applications due to its unique advantages in describing the hereditary and memory properties of multifarious materials and processes [1–4]. As a powerful tool used to describe most of the real-world behaviors, fractional-order systems can provide more practical value and accurate results in practice [5–11], such as abnormal diffusion, heat conduction electronic, some biological systems, and viscoelastic systems. Sequentially, many researchers have paid close attention to the applications of fractional-order differential equations in both engineering and theory and have drawn some wonderful and meaningful results in the literature [12–17].As we know that it is difficult to build a precise physical model of the engineering plant because of the uncertainties and noises. Thus, the fractional-order nonlinear system control with uncertainties and disturbances is still a challenging and attractive research field. Due to the inherent approximation capability, neural networks (NNs) or fuzzy logic systems are usually used to approximate the system uncertainties in integer-order system. The scholars in [17] have designed an adaptive fuzzy control scheme for a class of commensurate fractional-order systems with parametric uncertainty and input constraint. In [18], an adaptive backstepping controller is designed for a class of commensurate fractional-order systems with unknown parameters based on the indirect Lyapunov method, in which the control problem of fractional order is converted to the integer-order one [19]. Using the fractional-order extension of the Lyapunov direct method, an adaptive backstepping control method for a class of commensurate fractional-order nonlinear systems with unknown nonlinearity is developed in [20]. In [21, 22], an output feedback control scheme for a class of triangular commensurate fractional-order nonlinear systems is given. For a class of commensurate fractional-order rotational mechanical system with disturbances and uncertainties, a robust adaptive NN control is presented in [23]. Based on dual radial basis function (RBF) NNs, an adaptive commensurate fractional sliding mode controller is proposed to enhance the performance of the system in [24]. In [25], an adaptive NN control scheme is given for a class of commensurate fractional-order systems with nonlinearities and backlash-like hysteresis. In [26], the consensus problem of commensurate fractional-order MIMO systems with linear models is researched via the observer-based protocols. For a class of commensurate uncertain fractional-order nonlinear systems with external disturbance and input saturation, an adaptive NN backstepping control method based on the indirect Lyapunov method is designed in [27]. In [28], the output tracking control problem of a class of incommensurate fractional-order nonlinear systems with unknown nonlinearities and external disturbance is researched, in which the indirect Lyapunov method is used to design the controller. For a class of incommensurate fractional-order nonlinear systems with external disturbances and input dead-zone, a neural network adaptive control scheme is developed in [29]. Although many adaptive control schemes for the uncertain fractional-order system have been considered in above algorithms, the fuzzy backstepping control method for the incommensurate fractional-order systems with input nonlinearity is still short of results.Input nonlinearity is another important topic when designing the controller, which needs to be considered [30]. In fact, the real actuators cannot be a perfectly good linear characteristic in all operating domains due to physical constraints [31]. Therefore, the nonlinearities of practical actuators can cause severe degradation to the system performance since the theoretical controller is designed from an ideal assumption, which may cause system damage and even put risk on human users [32]. Input saturation often occurs in practical engineering, which is a source of limiting the system performance severely and even leads to the system instability. The study of control input saturation has become increasingly popular in recent years [33–37]. To solve the problem of actuator saturation, two methods are often used. One is to choose a low gain control law that the control input signal can be limited in a region to avoid saturation, and the other is to estimate the saturation region in the presence of actuator saturation. Generally speaking, it is very challenging to design a controller for the nonlinear uncertain systems with the input saturation problem.Therefore, the problem that nonlinear fractional-order systems with input saturation must be cope with when designing and implementing the control systems. Meanwhile, many real systems are better described by incommensurate fractional-order nonlinear systems, such as lithium-ion battery dynamics [38, 39], ultracapacitor dynamics [40], and quadratic boost converter [41], which means that the order of the fractional-order nonlinear systems should be extended to incommensurate one for the controller design. Motivated by the aforementioned observations, an adaptive fuzzy control method for the incommensurate fractional-order nonlinear uncertain systems with external disturbance and input saturation is presented in this study. The contributions are summarized as follows:(1)
Different from [27], the tracking controller is designed for a class of incommensurate fractional-order nonlinear uncertain system with external disturbance and input saturation, in which the indirect Lyapunov method with frequency distributed model is used to analyze the stability.(2)
Different from [25, 27], three different fractional-order adaptation laws are designed to update the fuzzy parameters from fuzzy logic system, unknown model constants, and the unknown bound from disturbances, respectively. Thus, the proposed method can handle different situations at the same time.(3)
Unlike the previous works [24–27], the orders of the parameters adaptation laws are not fixed the order of the fractional-order nonlinear system in this research, which provides more degree of freedom.This paper is organized as follows: Section2 introduces some preliminaries. Section 3 presents the adaptive fractional-order controller design and stability analysis. Section 4 provides the simulation results to illustrate the proposed controller. Section 5 concludes this article.


## 2. Preliminary

Theα‐th Caputo fractional derivative is defined as follows [42]:(1)t0Dtαft=1Γn−α∫t0tfntt−τα+1−ndτ,where Γα=∫0∞e−ttα−1dt, n−1<α<n,n∈ℤ+ and t0Dtα is the classical α−th order derivative operator. When t0=0, t0Dtα can be abbreviated as Dα.Lemma 1 (see [43]).
Consider a nonlinear fractional-order system(2)Dαxt=fxt,α∈0,1,xt∈ℝn,the system is exactly equivalent to the continuous frequency distributed model described by(3)∂zω,t∂t=−ωzω,t+fxt,xt=∫0∞μαωzω,tdω,where μαω=sinαπ/ωαπ. zω,t is the infinite dimension distributed state variable.In the developed control design procedure, fuzzy logic system will be used to approximate any continuous functionfX on a compact set Ω.Lemma 2 (see [44]).
The unknown smooth nonlinear functionfX can be approximated by a fuzzy logic system W∗Tϑx as follows:(4)fx=W∗Tϑx+ε,∀ε>0,where W=W1W2⋯WN∈ℝN is the weight vector, ϑx=ϑ1x,ϑ2x,…,ϑNxT is the fuzzy basis function vector, N is the number of inference rules, and ε represents the optimal approximation error. The optimal value W∗ of the parameter W is given by(5)W∗=argminWsupXfx−WTϑx.


## 3. Adaptive Fuzzy Backstepping Controller

In this paper, we consider a class of incommensurate fractional-order systems presented as follows:(6)Dαixi=xi+1+fix¯i+ΨiTx¯iθi+ζi,i=1,2,…,n−1,Dαnxn=buν+fnx+ΨnTxθn+ζn,y=x1,where αi∈0,1 is the system incommensurate fractional order, x=x1x2⋯xnT∈ℝn and x¯i=x1x2⋯xiT∈ℝi are the state vectors, i=1,2,…,n. fi⋅∈ℝ and Ψi⋅∈ℝpi denote the unknown parts and known parts of nonlinear functions, respectively. θi∈ℝpi is the unknown constant vector, ζi∈ℝ is the unknown disturbance, b∈ℝ is a known constant, and y∈ℝ is the system output. Control input uν expresses an input saturation defined as follows [45]:(7)u=satν=umax,ν≥umax,ν,umin<ν<umax,umin,ν≤umin,where ν is the input signal of the input saturation nonlinearity and umax>0 and umin<0 are the unknown constants.For (7), the input saturation function can be approximated by a smooth piecewise function defined as follows:(8)u=satν=gν+Δν,where(9)gν=umax⋅tanhνumax,ν≥0,umin⋅tanhνumin,ν<0,=umax⋅eν/umax−e−ν/umaxeν/umax+e−ν/umax,ν≥0,umin⋅eν/umin−e−ν/umineν/umin+e−ν/umin,ν<0,and Δν=satν−gν, satisfying(10)Δν=satν−gν≤maxumax1−tanh1,umintanh1−1=D.With the theorem of the mean, there exists a constantμ,0<μ<1 to make(11)gν=gν0+gνμν−ν0,where(12)gνμ=∂gν∂νν=νμ,νμ=μν+1−μν0.When selectingν0=0, (11) can be represented as(13)gν=gνμν.By substituting (8) and (13) into (6), system (6) can be rewritten as(14)Dαixi=xi+1+fix¯i+ΨiTx¯iθi+ζi,i=1,2,…,n−1,Dαnxn=bgνμν+fnx+ΨnTxθn+bΔν+ζn,y=x1.Our target is to design a inputν such that the system output y can follow the desired signal yd. Some following assumptions for the controller design are given.Assumption 1.
It is supposed that the reference signalsyd and the nth order derivatives Dnyd are continuous and bounded.Assumption 2.
There existsζ¯i>0, such that ζi≤ζ¯i,ζ¯i≥0.
In order to design a fractional-order backstepping algorithm for fractional-order nonlinear system (14), a virtual control law is found for every subsystem. Finally, all the system equations are considered and the control law is established to guarantee the closed-loop system stability. The error equation is defined as follows:(15)e1=y−yd,ei=xi−υi−1,i=2,…n,where υi−1 is the virtual control input. The virtual control input υi−1 and controller ν are designed as follows:(16)υ1=−W^1Tϑ1x¯1−k1e1−ξ^1signe1−Ψ1Tx¯1θ^1+Dα1yd,υi=−W^iTϑix¯i−kiei−ξ^isignei−ΨiTx¯iθ^i−ei−1,i=2,…,n−1,ν=1bgνμ−W^nTϑnx¯n−knen−ξ^nsignen−ΨnTx¯nθ^n−en−1,where θ^i is the estimation of the unknown constant vector θi (i=1,2,…,n). ki>0 is the design parameter. ξ^i is the estimation of the unknown constant ξi defined as follows:(17)ξi=ζ¯i+ε¯i,i=1,2,…,n−1,ξn=ζ¯n+ε¯n+bD,where ε¯i≥εix¯i,εix¯i=F^ix¯i,Wi∗−Fix¯i.
W
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i is the fuzzy logic system used to approximate the unknown function Fix¯i, where(18)F1x¯1=f1x¯1−Dα1yd,Fix¯i=fix¯i−Dαiυi−1,i=2,…,n,where W^i∈ℝmi is the parameter estimation of the ideal parameter Wi∗, which is described by(19)Wi∗=argminWisupx¯i∈ℝiFix¯i−F^ix¯i,Wi.
The fractional-order adaptation laws are designed as follows:(20)DβiW^i=σiϑix¯iei,Dγiθ^i=ηiΨix¯iei,Dχiξ^i=ρiei,where βi,γi,χi∈0,1, and σi,ηi,ρi>0 are the design parameters.Theorem 1.
Consider system (6) with input saturation, if the control input and virtual control inputs are chosen as (16), and the adaptation laws are designed as (20), then all the signals in the closed-loop system are globally uniformly bounded with the proper design parameters ki,βi,γi,χi,σi,ηi and ρi,i=1,2,…,n, and the tracking error e1=y−yd tend to zero asymptotically when t⟶∞.Proof.Step 1.
Letθ˜1=θ1−θ^1, then the following equation can be obtained:(21)Dγ1θ˜1=Dγ1θ1−Dγ1θ^1=−Dγ1θ^1.
Based on Lemma1, (21) will be(22)∂zθ˜1ω,t∂t=−ωzθ˜1ω,t−Dγ1θ^1,θ˜1=∫0∞μγ1ωzθ˜1ω,tdω,where zθ˜1ω,t∈ℝp1 and μγ1ω=sinγ1π/ωγ1π.
Define(23)W˜i=Wi∗−W^i.
Based on Lemma2 and (19), one can obtain(24)f^1x¯1,W^1−f1x¯1=f^1x¯1,W^1−f^1x¯1,W1∗+f^1x¯1,W1∗−f1x¯1=W^1Tϑ1x¯1−W1∗Tϑ1x¯1+ε1x¯1=−W˜1Tϑ1x¯1+ε1x¯1.
Due to (1) and the estimated error W˜1=W1∗−W^1 from (23), the following equation can be obtained:(25)Dβ1W˜1=Dβ1W1∗−Dβ1W^1=−Dβ1W^1.
According to Lemma1 and (25), the following frequency distributed model can be obtained:(26)∂zW˜1ω,t∂t=−ωzW˜1ω,t−Dβ1W^1,W˜1=∫0∞μβ1ωzW˜1ω,tdω,where zW˜1ω,t∈ℝm1 and μβ1ω=sinβ1π/ωβ1π.
On the other hand, it follows from (14), (24), and error e1=y−yd that(27)Dα1e1=Dα1x1−Dα1yd=x2+f1x¯1+Ψ1Tx¯1θ1+ζ1−Dα1yd=x2+F1x¯1+Ψ1Tx¯1θ1+ζ1=x2+F1x¯1−F^1x¯1,W^1+F^1x¯1,W^1+Ψ1Tx¯1θ1+ζ1=x2+W˜1Tϑ1x¯1+W^1Tϑ1x¯1−ε1x¯1+Ψ1Tx¯1θ1+ζ1.
Substituting virtual control inputυ1 and error equation e2=x2−υ1 into (27) gives(28)Dα1e1=e2−k1e1−ξ^1signe1+W˜1Tϑ1x¯1+Ψ1Tx¯1θ˜1−ε1x¯1+ζ1.
According to Lemma1, equation (28) will be(29)∂z1ω,t∂t=−ωz1ω,t+e2−k1e1−ξ^1signe1+W˜1Tϑ1x¯1+Ψ1Tx¯1θ^1−ε1x¯1+ζ1,e1=∫0∞μα1ωz1ω,tdω,where z1ω,t∈ℝ and μα1ω=sinα1π/ωα1π.
Letξ˜1=ξ1−ξ^1, then the following equation can be obtained:(30)Dχ1ξ˜1=Dχ1ξ1−Dχ1ξ^1=−Dχ1ξ^1.
Based on Lemma1, (30) will be(31)∂zξ˜1ω,t∂t=−ωzξ˜1ω,t−Dχ1ξ^1,ξ˜1=∫0∞μχ1ωzξ˜1ω,tdω,where zξ˜1ω,t∈ℝ and μχ1ω=sinχ1π/ωχ1π.
Selecting the Lyapunov functionV1 as(32)V1=12∫0∞μα1ωz12ω,tdω+12σ1∫0∞μβ1ωzW˜1Tω,tzW˜1ω,tdω+12η1∫0∞μγ1ωzθ˜1Tω,tzθ˜1ω,tdω+12ρ1∫0∞μχ1ωzξ˜12ω,tdω.
Based on frequency distributed models (22), (26), (29), and (31), the derivative of V1 is expressed as follows:(33)V˙1=−1σ1∫0∞ωμβ1ωzW˜1Tω,tzW˜1ω,tdω−1η1∫0∞ωμγ1ωzθ˜1Tω,tzθ˜1ω,tdω−∫0∞ωμα1ωz12ω,tdω−1ρ1∫0∞ωμχ1ωzξ˜1ω,tzξ˜1ω,tdω−W˜1T1σ1Dβ1W^1−e1ϑ1x¯1−θ˜1T1η1Dγ1θ^1−e1Ψ1x¯1−1ρ1ξ˜1Dχ1ξ^1+e1e2−k1e12−ξ^1e1−ε1x¯1e1+ζ1e1.
According to the fractional-order adaptation laws (20), equation (33) can be rewritten as follows:(34)V˙1=−1σ1∫0∞ωμβ1ωzW˜1Tω,tzW˜1ω,tdω−1η1∫0∞ωμγ1ωzθ˜1Tω,tzθ˜1ω,tdω−∫0∞ωμα1ωz12ω,tdω−1ρ1∫0∞ωμχ1ωzξ˜1ω,tzξ˜1ω,tdω−1ρ1ξ˜1Dχ1ξ^1+e1e2−k1e12−ξ^1e1−ε1x¯1e1+ζ1e1≤−1σ1∫0∞ωμβ1ωzW˜1Tω,tzW˜1ω,tdω−1η1∫0∞ωμγ1ωzθ˜1Tω,tzθ˜1ω,tdω−∫0∞ωμα1ωz12ω,tdω−1ρ1∫0∞ωμχ1ωzξ˜1ω,tzξ˜1ω,tdω−1ρ1ξ˜1Dχ1ξ^1+e1e2−k1e12−ξ^1−ε¯1−ζ¯1e1≤−1σ1∫0∞ωμβ1ωzW˜1Tω,tzW˜1ω,tdω−1η1∫0∞ωμγ1ωzθ˜1Tω,tzθ˜1ω,tdω−∫0∞ωμα1ωz12ω,tdω−1ρ1∫0∞ωμχ1ωzξ˜1ω,tzξ˜1ω,tdω+e1e2−k1e12−ξ˜11ρ1Dχ1ξ^1−e1.
Based on the fractional-order adaptation laws (20) and in equation (34), if e2=0 and k1>0, one can obtain V˙1<0.Step 2.
Letθ˜2=θ2−θ^2, then the following equation can be obtained:(35)Dγ2θ˜2=Dγ2θ2−Dγ2θ^2=−Dγ2θ^2.
Based on Lemma1, (35) will be(36)∂zθ˜2ω,t∂t=−ωzθ˜2ω,t−Dγ2θ^2,θ˜2=∫0∞μγ2ωzθ˜2ω,tdω,where zθ˜2ω,t∈ℝp2 and μγ2ω=sinγ2π/ωγ2π.
Based on the fuzzy logic system, it follows from the fractional-order system (14), error equation (15), and unknown function (18) that(37)Dα2e2=Dα2x2−Dα2υ1=x3+f2x¯2+Ψ2Tx¯2θ2+ζ2−Dα2υ1=x3+F2x¯2+Ψ2Tx¯2θ2+ζ2=x3+W˜2Tϑ2x¯2+W^2Tϑ2x¯2−ε2x¯2+Ψ2Tx¯2θ2+ζ2,where W˜2=W2∗−W^2 is the estimated error. Then, the following equation can be obtained:(38)Dβ2W˜2=Dβ2W2∗−Dβ2W^2=−Dβ2W^2.
According to Lemma1 and (38), the following frequency distributed model can be obtained:(39)∂zW˜2ω,t∂t=−ωzW˜2ω,t−Dβ2W^2,W˜2=∫0∞μβ2ωzW˜2ω,tdω,where zW˜2ω,t∈m2 and μβ2ω=sinβ2π/ωβ2π.
Substituting virtual control inputυ2 and error equation e3=x3−υ2 into (37) gives(40)Dα2e2=e3−k2e2−ξ^2signe2+W˜2Tϑ2x¯2+Ψ2Tx¯2θ˜2−e1−ε2x¯2+ζ2.
Its frequency distributed model corresponds to(41)∂z2ω,t∂t=−ωz2ω,t+e3−k2e2−ξ^2signe2+W˜2Tϑ2x¯2+Ψ2Tx¯2θ˜2−e1−ε2x¯2+ζ2,e2=∫0∞μα2ωz2ω,tdω,where z2ω,t∈ℝ and μα2ω=sinα2π/ωα2π.
Letξ˜2=ξ2∗−ξ^2, then the following equation can be obtained:(42)Dχ2ξ˜2=Dχ2ξ2∗−Dχ2ξ^2=−Dχ2ξ^2.
According to Lemma1, (42) will be(43)∂zξ˜2ω,t∂t=−ωzξ˜2ω,t−Dχ2ξ^2,ξ˜2=∫0∞μχ2ωzξ˜2ω,tdω,where zξ˜2ω,t∈ and μχ2ω=sinχ2π/ωχ2π.
Selecting the Lyapunov functionV2 as(44)V2=V1+12∫0∞μα2ωz22ω,tdω+12σ2∫0∞μβ2ωzW˜2Tω,tzW˜2ω,tdω+12η2∫0∞μγ2ωzθ˜2Tω,tzθ˜2ω,tdω+12ρ2∫0∞μχ2ωzξ˜22ω,tdω.
According to frequency distributed models (36), (39), (41), and (43), the derivative of V2 is expressed as(45)V˙2=V˙1−1σ2∫0∞ωμβ2ωzW˜2Tω,tzW˜2ω,tdω−1η2∫0∞ωμγ2ωzθ˜2Tω,tzθ˜2ω,tdω−∫0∞ωμα2ωz22ω,tdω−1ρ2∫0∞ωμχ2ωzξ˜2ω,tzξ˜2ω,tdω−W˜2T1σ2Dβ2W^2−e2ϑ2x¯2−θ˜2T1η2Dγ2θ^2−e2Ψ2x¯2−1ρ2ξ˜2Dχ2ξ^2+e2e3−k2e22−e1e2−ξ^2e2−ε2x¯2e2+ζ2e2≤V˙1−1σ2∫0∞ωμβ2ωzW˜2Tω,tzW˜2ω,tdω−1η2∫0∞ωμγ2ωzθ˜2Tω,tzθ˜2ω,tdω−∫0∞ωμα2ωz22ω,tdω−1ρ2∫0∞ωμχ2ωzξ˜2ω,tzξ˜2ω,tdω−W˜2T1σ2Dβ2W^2−e2ϑ2x¯2−θ˜2T1η2Dγ2θ^2−e2Ψ2x¯2−ξ˜21ρ2Dχ2ξ^2−e2+e2e3−k2e22−e1e2≤−1σ1∫0∞ωμβ1ωzW˜1Tω,tzW˜1ω,tdω−1η1∫0∞ωμγ1ωzθ˜1Tω,tzθ˜1ω,tdω−1ρ1∫0∞ωμχ1ωzξ˜1ω,tzξ˜1ω,tdω−∫0∞ωμα1ωz12ω,tdω−ξ˜11ρ1Dχ1ξ^1−e1−1σ2∫0∞ωμβ2ωzW˜2Tω,tzW˜2ω,tdω−1η2∫0∞ωμγ2ωzθ˜2Tω,tzθ˜2ω,tdω−∫0∞ωμα2ωz22ω,tdω−1ρ2∫0∞ωμχ2ωzξ˜2ω,tzξ˜2ω,tdω−W˜2T1σ2Dβ2W^2−e2ϑ2x¯2−θ˜2T1η2Dγ2θ^2−e2Ψ2x¯2−ξ˜21ρ2Dχ2ξ^2−e2−k1e12−k2e22+e2e3.
Based on fractional-order adaptation laws (20) and in equation (45), if e3=0 and k2>0, one can get V˙2<0.Stepi,3≤i≤n−1.
Based on the fuzzy logic system, it follows from the fractional-order system (14), error equation (15), and unknown function (18) that(46)Dαiei=Dαixi−Dαiυi−1=xi+1+fix¯i+ΨiTx¯iθi+ζi−Dαiυi−1=xi+1+Fix¯i+ΨiTx¯iθi+ζi=xi+1+W˜iTϑix¯i+W^iTϑix¯i−εix¯i+ΨiTx¯iθi+ζi,where W˜i=Wi∗−W^i is the estimated error for the fuzzy logic system. Then, the following equation can be obtained:(47)DβiW˜i=DβiWi∗−DβiW^i=−DβiW^i.
According to Lemma1 and (47), the following frequency distributed model can be obtained:(48)∂zW˜iω,t∂t=−ωzW˜iω,t−DβiW^i,W˜i=∫0∞μβiωzW˜iω,tdω,where zW˜iω,t∈ℝmi and μβiω=sinβiπ/ωβiπ.
Substituting virtual control inputυi and error equation ei+1=xi+1−υi into (46) gives(49)Dαiei=ei+1−kiei−ξ^isignei+W˜iTϑix¯i+ΨiTx¯iθ˜i−ei−1−εix¯i+ζi.
Its frequency distributed model corresponds to(50)∂ziω,t∂t=−ωziω,t+ei+1−kiei−ξ^isignei+W˜iTϑix¯i+ΨiTx¯iθ˜i−ei−1−εix¯i+ζi,ei=∫0∞μαiωziω,tdω,where ziω,t∈ℝ and μαiω=sinαiπ/ωαiπ. θ˜i=θi−θ^i is parameter estimation error, and the following equation can be obtained:(51)Dγiθ˜i=Dγiθi−Dγiθ^i=−Dγiθ^i.
Based on Lemma1, (51) will be(52)∂zθ˜iω,t∂t=−ωzθ˜iω,t−Dγiθ^i,θ˜i=∫0∞μγiωzθ˜iω,tdω,where zθ˜iω,t∈ℝpi and μγiω=sinγiπ/ωγiπ.
Letξ˜i=ξi∗−ξ^i, then the following equation can be obtained:(53)Dχiξ˜i=Dχiξi∗−Dχiξ^i=−Dχiξ^i.
According to Lemma1, (53) will be(54)∂zξ˜iω,t∂t=−ωzξ˜iω,t−Dχiξ^i,ξ˜i=∫0∞μχiωzξ˜iω,tdω,where zξ˜iω,t∈ℝ and μχiω=sinχiπ/ωχiπ.
Selecting the Lyapunov functionVi as(55)Vi=Vi−1+12∫0∞μαiωzi2ω,tdω+12σi∫0∞μβiωzW˜iTω,tzW˜iω,tdω+12ηi∫0∞μγiωzθ˜iTω,tzθ˜iω,tdω+12ρi∫0∞μχiωzξ˜i2ω,tdω.
Then, its derivative on the basis of frequency distributed models (48), (50), (52), and (54) is expressed as(56)V˙i=V˙i−1−∫0∞ωμαiωzi2ω,tdω−1σi∫0∞ωμβiωzW˜iTω,tzW˜iω,tdω−1ηi∫0∞ωμγiωzθ˜iTω,tzθ˜iω,tdω−1ρi∫0∞ωμχiωzξ˜i2ω,tdω−W˜iT1σiDβiW^i−ϑix¯iei−θ˜iT1ηiDγiθ^i−ΨiTx¯iei−ξ˜i1ρiDχiξ^i−kiei2+eiei+1−ei−1ei−ξ^iei−εix¯i−ζiei≤V˙i−1−∫0∞ωμαiωzi2ω,tdω−1σi∫0∞ωμβiωzW˜iTω,tzW˜iω,tdω−1ηi∫0∞ωμγiωzθ˜iTω,tzθ˜iω,tdω−1ρi∫0∞ωμχiωzξ˜i2ω,tdω−W˜iT1σiDβiW^i−ϑix¯iei−θ˜iT1ηiDγiθ^i−ΨiTx¯iei−ξ˜i1ρiDχiξ^i−ei−kiei2+eiei+1−ei−1ei≤−∑l=1i∫0∞ωμαlωzl2ω,tdω−∑l=1i1σl∫0∞ωμβlωzW˜lTω,tzW˜iω,tdω−∑l=1i1ηl∫0∞ωμγlωzθ˜lTω,tzθ˜iω,tdω−∑l=1i1ρl∫0∞ωμχlωzξ˜l2ω,tdω−∑l=1iW˜lT1σlDβlW^l−ϑlx¯lel−∑l=1iθ˜lT1ηlDγlθ^l−ΨlTx¯lel−∑l=1iξ˜l1ρlDχlξ^l−el−∑l=1iklel2+eiei+1.
According to fractional-order adaptation laws (20) and in equation (56), if ei+1=0 and ki>0, one can get V˙i<0.Stepn.
Based on error equationen=xn−υn−1, fractional-order system (14), and unknown function (18), one has(57)Dαnen=bgνμν+fnx+ΨnTxθn+bΔν+ζn−Dαnυn−1=bgνμν+Fnx+ΨnTxθn+bΔν+ζn=bgνμν+W˜nTϑnx+W^nTϑnx+ΨnTxθn−εnx¯n+bΔν+ζn,where W˜n=Wn∗−W^n is the estimated error for the fuzzy logic system, and the following equation can be obtained:(58)DβnW˜n=DβnWn∗−DβnW^n=−DβnW^n.
Due to Lemma1, (58) will be(59)∂zW˜nω,t∂t=−ωzW˜nω,t−DβnW^n,W˜n=∫0∞μβnωzW˜nω,tdω,where zW˜nω,t∈ℝmn and μβnω=sinβnπ/ωβnπ.
Substituting the designed controllerν in (16) into (57) gives(60)Dαnen=−knen−ξ^nsignen−en−1+W˜nTϑnx+ΨnTxθ˜n−εnx+bΔν+ζn,then the following frequency distributed model is obtained:(61)∂znω,t∂t=−knen−ξ^nsignen−en−1+W˜nTϑnx+ΨnTxθ˜n−εnx+bΔν+ζn,en=∫0∞μαnωznω,tdω,where znω,t∈ℝ and μαnω=sinαnπ/ωαnπ. θ˜n=θn−θ^n is parameter estimation error, and the following equation can be obtained:(62)Dγnθ˜n=Dγnθn−Dγnθ^n=−Dγnθ^n.
Based on Lemma1, (62) will be(63)∂zθ˜nω,t∂t=−ωzθ˜nω,t−Dγnθ^n,θ˜n=∫0∞μγnωzθ˜nω,tdω,where zθ˜nω,t∈ℝpn and μγnω=sinγnπ/ωγnπ.
Letξ˜n=ξn−ξ^n, then the following equation can be obtained:(64)Dχnξ˜n=Dχnξn−Dχnξ^n=−Dχnξ^n.
According to Lemma1, (64) will be(65)∂zξ˜nω,t∂t=−ωzξ˜nω,t−Dχnξ^n,ξ˜n=∫0∞μχnωzξ˜nω,tdω,where zξ˜nω,t∈ℝ and μχnω=sinχnπ/ωχnπ.
Selecting the Lyapunov functionVn as(66)Vn=Vn−1+12∫0∞μαnωzn2ω,tdω+12σn∫0∞μβnωzW˜nTω,tzW˜nω,tdω+12ηn∫0∞μγnωzθ˜nTω,tzθ˜nω,tdω+12ρn∫0∞μχnωzξ˜n2ω,tdω.
Based on the procedures in stepi,3≤i≤n−1, the derivative of Vn on the basis of frequency distributed models (59), (61), (63), and (65) is expressed as follows:(67)V˙n=V˙n−1−∫0∞ωμαnωzn2ω,tdω−1σn∫0∞ωμβnωzW˜nTω,tzW˜nω,tdω−1ηn∫0∞ωμγnωzθ˜nTω,tzθ˜nω,tdω−1ρn∫0∞ωμχnωzξ˜n2ω,tdω−W˜nT1σnDβnW^n−ϑnx¯nen−θ˜nT1ηnDγnθ^n−ΨnTx¯nen−ξ˜n1ρnDχnξ^n−knen2−en−1en−ξ^nen−εnx¯n−bΔν−ζnen≤V˙n−1−∫0∞ωμαnωzn2ω,tdω−1σn∫0∞ωμβnωzW˜nTω,tzW˜nω,tdω−1ηn∫0∞ωμγnωzθ˜nTω,tzθ˜nω,tdω−1ρn∫0∞ωμχnωzξ˜n2ω,tdω−W˜nT1σnDβnW^n−ϑnx¯nen−θ˜nT1ηnDγnθ^n−ΨnTx¯nen−ξ˜n1ρnDχnξ^n−en−knen2−en−1en≤−∑l=1n∫0∞ωμαlωzl2ω,tdω−∑l=1n1σl∫0∞ωμβlωzW˜lTω,tzW˜iω,tdω−∑l=1n1ηl∫0∞ωμγlωzθ˜lTω,tzθ˜iω,tdω−∑l=1n1ρl∫0∞ωμχlωzξ˜l2ω,tdω−∑l=1nW˜lT1σlDβlW^l−ϑlx¯lel−∑l=1nθ˜lT1ηlDγlθ^l−ΨlTx¯lel−∑l=1nξ˜l1ρlDχlξ^l−el−∑l=1nklel2.
Based on fractional-order adaptation laws (20) and in equation (67), if kn>0, one can get V˙n<0.
According tostep1,2,i,n, if the control input is chosen as (16), and the adaptation laws are designed as (20) then with a proper choice of the design parameters ki,βi,γi,χi,σi,ηi and ρi,i=1,2,…,n, one can get V˙i<0,i=1,2,…,n. Due to the LaSalle invariance principle [46], ziω,t, zW˜iω,t, zθ˜iω,t, and zξ˜iω,t can be close to the set of all points. When V˙n=0, one can obtain ziω,t=0, zW˜iω,t=0, zθ˜iω,t=0, and zξ˜iω,t=0, which is the only equilibrium point. That is the error variables ei, W˜i, θ˜i, and ξ˜i convergent to zero asymptotically. Therefore, the tracking error e1 tends to zero asymptotically and all the signals are uniformly bounded.Remark 1.
An adaptive fuzzy backstepping algorithm for a class of fractional-order systems with external disturbances and input saturation is proposed in Theorem1. The fuzzy logic system is used to approximate the unknown nonlinear uncertainties including the fractional-order function Dα1yd,Dαiυi−1,i=2,…,n in each step of the backstepping, which may bring some computation burden of the control system for the parameter estimation to obtain accurate approximation effect.Remark 2.
There are the sign functions in the controller (16), which can make the chattering phenomenon for the closed-loop control system. However, the sign function used the controller (16) can ensure that the tracking error is bounded, and the chattering phenomenon can be alleviated by replacing the sign function by a continuous function, such as arctan10.Remark 3.
The proposed algorithm can be applied to both the incommensurate and commensurate fractional-order systems with input saturation, in which the orders of the parameter estimation laws (βi,γi and χi,i=1,2,…,n) cannot be fixed to the system order (αi,i=1,2,…,n). Therefore, the proposed algorithm brings more degree of freedom and better control performance.


## 4. Simulation

Two examples are presented in this simulation section to show the effectiveness of the proposed method.


### 4.1. Example 1

The following three-order incommensurate fractional-order nonlinear system is considered as(68)D0.8x1=x2−0.05x12+0.3x1θ1+0.1sint+0.3cost,D0.6x2=x3+x2−0.8x126+x14+sinx1x2θ2+0.1sint,D0.7x3=6uν−e−5x12x2sin3x2−cosx3θ3+0.06cost,where f1x¯1=−0.05x12, f2x¯2=x2−0.8x12/6+x14, and f3x¯3=−e−5x12x2sin3x2 are the unknown functions. Ψ1=0.3x1, Ψ2=sinx1x2, and Ψ3=−cosx3 are the known functions. θ1=0.1,θ2=0.15 and θ3=0.6 are the unknown constants. ζ1=0.1sint+0.3cost, ζ2=0.1sint, and ζ3=0.06cost are the unknown uncertain disturbances.The input saturationuν is presented as(69)uν=satν=25,ν≥25,ν,−25<ν<25,−25,ν≤−25.The reference signal for the system output is chosen asyd=0.85sint+1.5. Three fuzzy systems are used in the presented controller. The first fuzzy system using x1 as its input defines the Gaussian membership functions as selecting the Lyapunov function Vn as(70)exp−12x1−xk10.182,xk1∈0.5k1−2k1=1,2,…,6.The second fuzzy system usesx1 and x2 as its inputs. The membership functions for the input x1 are chosen as (70), and the membership functions for the input x2 are chosen as(71)exp−12x2−xk20.52,xk2∈k2−2k2=1,2,3.The third fuzzy system employsx1, x2, and x3 as its inputs. The membership functions for the input x1 and x2 are chosen as the same as the second one, and the membership functions for the input x3 are chosen as(72)exp−12x3−xk30.162,xk3∈0.5k3−1.5k3=1,2,…,5.The design parameters are chosen ask1=k2=k3=35, σ1=σ2=σ3=3, η1=η2=η3=0.0001, ρ1=ρ2=ρ3=0.0001, β1=β2=β3=0.6, γ1=0.8,γ2=0.6,γ3=0.7, χ1=0.8, χ2=0.85, and χ3=0.9. The initial conditions of parameter estimation are W^10=06×1,W^20=018×1,W^30=090×1, θ^10=θ^20=θ^30=0, and ξ^10=ξ^20=ξ^30=0.The simulation results are illustrated in Figures1–6. The tracking performance of the system output y in comparison with reference signal yd is shown in Figure 1. It demonstrates that the tracking error e1=y−yd has a rapid convergence with the unknown nonlinear terms, unknown parameters, and input saturation, implying a good tracking performance. However, due to the approximation error from unknown nonlinear terms approximated by fuzzy logic system and chattering phenomenon from the sign⋅ in controller, the tracking error e1 converges to a small range of zero. The trajectories of the system states x2 and x3 are indicated in Figure 2, and the norm of parameters estimation of the fuzzy logic system is shown in Figure 3. Figure 4 shows the estimation of the unknown constants θ^1,θ^2, and θ^3. Figure 5 displays the estimation of the ξ^1,ξ^2, and ξ^3, and system input saturation uν is presented in Figure 6. Meanwhile, Figures 2–6 demonstrate the boundedness of the signals in the closed-loop control system.Figure 1
Evolution ofy, yd, and tracking error e1.
(a)
(b)Figure 2
Evolution of statex2 and x3.Figure 3
Evolution of the norm of the fuzzy logic system parameters.Figure 4
Evolution of the estimation of the unknown constantsθ^1,θ^2, and θ^3.Figure 5
Evolution of the estimation of theξ^1,ξ^2, and ξ^3.Figure 6
Evolution of the control input.


### 4.2. Example 2

To show more results of the proposed method, the following incommensurate fractional-order nonlinear system is considered:(73)D0.95x1=x2+107x1−x13+cosx1θ2+0.01sint,D0.98x2=x3+10x1−x2+0.5x1x2θ1+0.05sint+cost,D0.96x3=5uν−1007x2+x2sinx3θ3+0.001cost,where f1x¯1=10/7x1−x13, f2x¯2=10x1−x2, and f3x¯3=−100/7x2 are the unknown functions. Ψ1=cosx1, Ψ2=0.5x1x2, and Ψ3=x2sinx3 are the known functions. θ1=0.3,θ2=0.5, and θ3=0.8 are the unknown constants. ζ1=0.01sint, ζ2=0.05sint+cost, and ζ3=0.001cost are the unknown uncertain disturbances.The input saturationuν is presented as(74)uν=satν=35,ν≥35,ν,−35<ν<35,−35,ν≤−35.The reference signal for the system output is chosen asyd=0.6cost+0.8. There are three fuzzy systems used in the presented controller. The first fuzzy system employs x1 as its input and defines the Gaussian membership functions as(75)exp−12x1−xk10.52,xk1∈1.5k1−6k1=1,2,…,6.The second fuzzy system usesx1 and x2 as its inputs. The membership functions for the input x1 are chosen as the same as (75), and the membership functions for the input x2 are chosen as(76)exp−12x2−xk21.12,xk2∈3k2−6k2=1,2,3.The third fuzzy system usesx2 as its inputs, and the membership functions are chosen as (76).The design parameters are chosen ask1=39,k2=k3=25, σ1=σ2=σ3=5, η1=η2=η3=0.0003, ρ1=ρ2=ρ3=0.0001, β1=β2=β3=0.6, γ1=0.5,γ2=0.8,γ3=0.7, χ1=0.8, χ2=0.6, and χ3=0.5. The initial conditions of parameter estimation are W^10=06×1,W^20=018×1,W^30=090×1, θ^10=θ^20=θ^30=0, and ξ^10=ξ^20=ξ^30=0.The tracking errore1=y−yd shown in Figure 7 converges to a smaller range of zero. Figure 8 shows that the trajectories of the system states x2 and x3 are bounded. The norm of parameters estimation of the fuzzy logic system and the estimation of the unknown constants θ^1,θ^2, and θ^3 are depicted in Figures 9 and 10. Figure 11 depicts the estimation of the unknown constants ξ^1,ξ^2, and ξ^3, and the input saturation uν is depicted in Figure 12, which is bounded.Figure 7
Evolution ofy, yd, and tracking error e1.
(a)
(b)Figure 8
Evolution of statex2 and x3.Figure 9
Evolution of the norm of the fuzzy logic system parameters.Figure 10
Evolution of the estimation of the unknown constantsθ^1,θ^2, and θ^3.Figure 11
Evolution of the estimation of theξ^1,ξ^2, and ξ^3.Figure 12
Evolution of the control input.Above simulation results demonstrate that the system output can track the reference signal in a small range of zero, and all the closed-loop signals are bounded, demonstrating the effectiveness of the proposed control scheme.


## 4.1. Example 1

The following three-order incommensurate fractional-order nonlinear system is considered as(68)D0.8x1=x2−0.05x12+0.3x1θ1+0.1sint+0.3cost,D0.6x2=x3+x2−0.8x126+x14+sinx1x2θ2+0.1sint,D0.7x3=6uν−e−5x12x2sin3x2−cosx3θ3+0.06cost,where f1x¯1=−0.05x12, f2x¯2=x2−0.8x12/6+x14, and f3x¯3=−e−5x12x2sin3x2 are the unknown functions. Ψ1=0.3x1, Ψ2=sinx1x2, and Ψ3=−cosx3 are the known functions. θ1=0.1,θ2=0.15 and θ3=0.6 are the unknown constants. ζ1=0.1sint+0.3cost, ζ2=0.1sint, and ζ3=0.06cost are the unknown uncertain disturbances.The input saturationuν is presented as(69)uν=satν=25,ν≥25,ν,−25<ν<25,−25,ν≤−25.The reference signal for the system output is chosen asyd=0.85sint+1.5. Three fuzzy systems are used in the presented controller. The first fuzzy system using x1 as its input defines the Gaussian membership functions as selecting the Lyapunov function Vn as(70)exp−12x1−xk10.182,xk1∈0.5k1−2k1=1,2,…,6.The second fuzzy system usesx1 and x2 as its inputs. The membership functions for the input x1 are chosen as (70), and the membership functions for the input x2 are chosen as(71)exp−12x2−xk20.52,xk2∈k2−2k2=1,2,3.The third fuzzy system employsx1, x2, and x3 as its inputs. The membership functions for the input x1 and x2 are chosen as the same as the second one, and the membership functions for the input x3 are chosen as(72)exp−12x3−xk30.162,xk3∈0.5k3−1.5k3=1,2,…,5.The design parameters are chosen ask1=k2=k3=35, σ1=σ2=σ3=3, η1=η2=η3=0.0001, ρ1=ρ2=ρ3=0.0001, β1=β2=β3=0.6, γ1=0.8,γ2=0.6,γ3=0.7, χ1=0.8, χ2=0.85, and χ3=0.9. The initial conditions of parameter estimation are W^10=06×1,W^20=018×1,W^30=090×1, θ^10=θ^20=θ^30=0, and ξ^10=ξ^20=ξ^30=0.The simulation results are illustrated in Figures1–6. The tracking performance of the system output y in comparison with reference signal yd is shown in Figure 1. It demonstrates that the tracking error e1=y−yd has a rapid convergence with the unknown nonlinear terms, unknown parameters, and input saturation, implying a good tracking performance. However, due to the approximation error from unknown nonlinear terms approximated by fuzzy logic system and chattering phenomenon from the sign⋅ in controller, the tracking error e1 converges to a small range of zero. The trajectories of the system states x2 and x3 are indicated in Figure 2, and the norm of parameters estimation of the fuzzy logic system is shown in Figure 3. Figure 4 shows the estimation of the unknown constants θ^1,θ^2, and θ^3. Figure 5 displays the estimation of the ξ^1,ξ^2, and ξ^3, and system input saturation uν is presented in Figure 6. Meanwhile, Figures 2–6 demonstrate the boundedness of the signals in the closed-loop control system.Figure 1
Evolution ofy, yd, and tracking error e1.
(a)
(b)Figure 2
Evolution of statex2 and x3.Figure 3
Evolution of the norm of the fuzzy logic system parameters.Figure 4
Evolution of the estimation of the unknown constantsθ^1,θ^2, and θ^3.Figure 5
Evolution of the estimation of theξ^1,ξ^2, and ξ^3.Figure 6
Evolution of the control input.


## 4.2. Example 2

To show more results of the proposed method, the following incommensurate fractional-order nonlinear system is considered:(73)D0.95x1=x2+107x1−x13+cosx1θ2+0.01sint,D0.98x2=x3+10x1−x2+0.5x1x2θ1+0.05sint+cost,D0.96x3=5uν−1007x2+x2sinx3θ3+0.001cost,where f1x¯1=10/7x1−x13, f2x¯2=10x1−x2, and f3x¯3=−100/7x2 are the unknown functions. Ψ1=cosx1, Ψ2=0.5x1x2, and Ψ3=x2sinx3 are the known functions. θ1=0.3,θ2=0.5, and θ3=0.8 are the unknown constants. ζ1=0.01sint, ζ2=0.05sint+cost, and ζ3=0.001cost are the unknown uncertain disturbances.The input saturationuν is presented as(74)uν=satν=35,ν≥35,ν,−35<ν<35,−35,ν≤−35.The reference signal for the system output is chosen asyd=0.6cost+0.8. There are three fuzzy systems used in the presented controller. The first fuzzy system employs x1 as its input and defines the Gaussian membership functions as(75)exp−12x1−xk10.52,xk1∈1.5k1−6k1=1,2,…,6.The second fuzzy system usesx1 and x2 as its inputs. The membership functions for the input x1 are chosen as the same as (75), and the membership functions for the input x2 are chosen as(76)exp−12x2−xk21.12,xk2∈3k2−6k2=1,2,3.The third fuzzy system usesx2 as its inputs, and the membership functions are chosen as (76).The design parameters are chosen ask1=39,k2=k3=25, σ1=σ2=σ3=5, η1=η2=η3=0.0003, ρ1=ρ2=ρ3=0.0001, β1=β2=β3=0.6, γ1=0.5,γ2=0.8,γ3=0.7, χ1=0.8, χ2=0.6, and χ3=0.5. The initial conditions of parameter estimation are W^10=06×1,W^20=018×1,W^30=090×1, θ^10=θ^20=θ^30=0, and ξ^10=ξ^20=ξ^30=0.The tracking errore1=y−yd shown in Figure 7 converges to a smaller range of zero. Figure 8 shows that the trajectories of the system states x2 and x3 are bounded. The norm of parameters estimation of the fuzzy logic system and the estimation of the unknown constants θ^1,θ^2, and θ^3 are depicted in Figures 9 and 10. Figure 11 depicts the estimation of the unknown constants ξ^1,ξ^2, and ξ^3, and the input saturation uν is depicted in Figure 12, which is bounded.Figure 7
Evolution ofy, yd, and tracking error e1.
(a)
(b)Figure 8
Evolution of statex2 and x3.Figure 9
Evolution of the norm of the fuzzy logic system parameters.Figure 10
Evolution of the estimation of the unknown constantsθ^1,θ^2, and θ^3.Figure 11
Evolution of the estimation of theξ^1,ξ^2, and ξ^3.Figure 12
Evolution of the control input.Above simulation results demonstrate that the system output can track the reference signal in a small range of zero, and all the closed-loop signals are bounded, demonstrating the effectiveness of the proposed control scheme.


## 5. Conclusion

In this paper, a new adaptive fuzzy backstepping control method for a class of incommensurate fractional-order nonlinear uncertain systems with external disturbance and input saturation is proposed, which can be considered as an extension of the commensurate one in [27]. In each step, the fuzzy logic system is used to approximate unknown nonlinear terms, and the fractional-order parameters update law as well as disturbance estimator are constructed based on the Lyapunov theorem for fractional-order systems with regard to the frequency distributed model. The proposed algorithm guarantees the convergence of the tracking error, and the orders of the adaptation laws are not fixed the order of the fractional-order nonlinear system, providing more degree of freedom compared with [24–27]. Two simulation examples are presented to demonstrate the effectiveness of proposed method, and the results show that the asymptotic tracking problem for a class of incommensurate fractional-order nonlinear uncertain systems with external disturbance and input saturation can be well solved by the proposed scheme. In the future, we will consider to look for suitable real system applications using the proposed method.



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*Source: 5643298-2019-12-20.xml*