# Composite Adaptive Fuzzy Prescribed Performance Control of Nonlinear Systems

Authors: Xudong Cao; Jianjun Wang; Wei Xiang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper proposes an adaptive fuzzy prescribed performance control (PPC) method of a class of uncertain nonlinear systems. Different from the traditional PPC approach that requires the exact values of the initial conditions, by using a new type of performance function, the proposed PPC scheme together with a composite adaptation law works effectively even without the knowledge of initial conditions. Meanwhile, the constructed disturbance observer and fuzzy logic systems can estimate system uncertainties including external disturbances and fuzzy approximation errors. Under the proposed tracking controller, the boundedness of all involved signals is guaranteed, and the tracking errors satisfy the prescribed performance bounds all the time. Finally, simulation results show the efficacy of the proposed method.


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## Body

## 1. Introduction

It is well known that the control performance of traditional control methods may not be satisfactory for uncertainties and external disturbances in real systems. In fact, most real-world systems’ model is very hard to be achieved, and it is very hard to implement controllers for these systems. Since the fuzzy logic system can be used to model nonlinear dynamics, it can be used to deal with the uncertainty of the system. At present, adaptive fuzzy control for uncertain systems has been widely studied [1–9]. Adaptive fuzzy control method is effective for tackling parametric uncertainties and external disturbances, in which tracking errors are utilized to update fuzzy parameters as well as to approximate unknown nonlinear functions. For fractional-order modified chaotic systems, linear control and fuzzy control are combined to realize synchronous control in [10]. Adaptive fuzzy impulsive control for nonlinear systems was presented in [11]. On the contrary, prescribed performance control (PPC) methods are proposed for nonlinear systems to get rapid stability of the controlled systems [12–16]. The PPC method usually includes two aspects: one is transforming the constrained system into an unconstrained equivalent transformation system, and the other is to prove that the transformed variable remains bounded. For example, in [13], PPC and backstepping technique were combined to explore a class of nontriangular structure nonlinear systems. Shao and Tong [14] investigated an effective PPC scheme for MIMO stochastic nonlinear systems. Usually, the traditional prescribed performance boundary (PPB) of tracking error e is described as(1)−λμt<e<μt,ife0>0,−μt<e<λμt,ife0<0,where λ ∈ [0, 1], μt=λ0−λ∞e−λ1t+λ∞, and λ0, λ∞, and λ1 are design parameters. If the initial value e(0) is unknown, the controller of PPC may not meet PPB (1), resulting in poor control performance. Recently, Bu et al. [17] proposed prescribed performance neural controllers of a flexible air-breathing hypersonic vehicle. By using the property of performance function, the problem of unknown initial value was solved, and the prescribed performance of the system state is guaranteed. Therefore, the performance function in [17] will be used to investigate the uncertain nonlinear system with dead-zone inputs.Learning ability is a fundamental feature of all intelligent behaviors. Up to now, there are many learning control methods which have been proposed for nonlinear systems. Iterative learning control technology is applicable to the repeated operation system on finite time under the premise that the initial value of the iteration error is zero [18–20]. After enough iterations, the controller with reasonable design parameters can realize the zero error of the system output/state over the entire operation interval. However, due to the limitation of the reset condition, there is an initial error in the actual situation. For various nonlinear uncertainties including the parameter uncertain system and the nonparametric uncertain system, designing learning controller based on Lyapunov synthesis method, solving the corresponding trajectory tracking problem, is a research hotspot in the field of learning control. To solve the problem of nonzero error initial value and to design the learning control system, the composite learning control method is proposed by Pan et al. [21, 22]. Based on comprehensive learning techniques, an adaptive dynamic surface control algorithm was proposed for integer-order systems with mismatched parameters in [21]. The interval integral of online recorded data was used to generate prediction error and instantaneous tracking error to update the parameter estimation. The composite learning technology can solve the shortcomings of the parallel learning adaptive control algorithm and effectively realize the parameter convergence under the condition of continuous excitation in the integer-order system. In [23, 24], a disturbance observer was proposed for nonlinear systems by means of the adaptive fuzzy composite learning control, where the PPC problem is not considered. In [25, 26], composite learning control was extended for fractional-order systems. To our knowledge, the composite learning PPC is rarely investigated.Inspired by the above works, this paper researches adaptive fuzzy PPC for nonlinear systems with unknown initial conditions. Firstly, the PPB conditions are given by the performance function. Then, the original system is transformed into an equivalent transformation system. Afterwards, an external disturbance observer is designed, which can estimate system uncertainties including external disturbances and fuzzy approximation errors. The proposed adaptive fuzzy composite learning law can not only guarantee the prescribed performance but also make accurate estimates of system uncertainties. The main contributions of this paper are summarized as follows. (1) The control method proposed in this paper can overcome the dependence of traditional PPC on initial values. (2) Disturbance observers constructed with prediction errors can accurately estimate system uncertainties. (3) Compared with the traditional adaptive fuzzy PPC method, the proposed method has better control performance.The rest of this work are arranged as follows. Some basic mathematical results and the problem statement are presented in Section2. The prescribed performance transformation, the adaptive fuzzy PPC design, and the observer design, as well as the stability analysis are included in Section 3. A simulation example is presented in Section 4 to show the validity of the proposed method, and some comparison results are also given. At last, Section 5 gives a sum-up of the whole research.


## 2. Preliminaries

Consider the following nonlinear system with dead-zone inputs(2)x˙=fx+dt+GxΓu,where x=x1,x2,…,xnT∈Rn is the state vector which is assumed to be measurable, fx=f1x,f2x,…,fnxT is an unknown smooth function, G(x) = diag(g1x, g2x, …, gnx) is a coefficient matrix, and gix is a known function, i = 1, 2, …, n. dt=d1t,d2t,…,dntT is an unknown external disturbance. u=u1,u2,…,unT is the controller output vector, and Γu=Γ1u1,Γ2u2,…,ΓnunT is the controller input vector subject to dead-zone type nonlinearity. Γi(ui) is described as(3)Γiui=mui−bri,ifui≥bri,0,if−bli<ui<bri,mui+bli,ifui≤−bli,where m, bri, and bli are positive constants. So, one can write (3) as(4)Γiui=mui+Δui,where(5)Δui=−mbir,ifui≥bir,−mui,if−bil<ui<bir,mbil,ifui≤−bil.According to (4), system (1) can be rewritten as(6)x˙=fx+mGxu+d¯t,where d¯t=d¯1t,d¯2t,…,d¯ntT and d¯it=dit+Δui.Definee=x−xd=e1,e2,…,enT and ei = xi − xdi, and xdi is the reference signal, i = 1, 2, …, n. The following assumptions need to be given:Assumption 1. The reference signal xdi and x˙di are known, and the initial xi(0) is unknown but boundedAssumption 2. The nonlinear function fi(x) is unknown but boundedAssumption 3. The gain function gix is known and gix≠0 for all xAssumption 4. The unknown external disturbance di(t) is time varying and satisfies dit≤l1i and d˙it≤l2i, where l1i and l2i are unknown positive constantsRemark 1.
It is well known that, in real-world systems, the initial conditions are very hard to be obtained, i.e., the initial conditions are unknown. In this paper, to solve this problem, based on above assumptions, this paper will design an adaptive fuzzy prescribed performance controller.Remark 2.
In this paper,fix¯ is assumed unknown smooth function, so fuzzy logic systems (FLSs) will be used to approximate it. If gix¯ is an uncertain smooth nonzero function, one can divide gix¯ into the nominal part g¯ix¯i and the perturbed part Δgi and incorporate Δgi into fix¯ and define f¯ix¯ as(7)f¯ix¯=fix¯+ΔgiΓiui,and then estimate f¯ix¯ by using FLSs. Therefore, this paper only considers the case that gix¯ is known smooth nonzero function.
Here, defineFi(x) = lifi(x), where li is a positive design constant and F^ix=θ^FiTφFix is the approximate of Fi(x) by using FLSs, where ϑ^Fi is an adjustable parameter vector and φFix is the fuzzy basis function vector. And Fix=θFi∗TφFix+εFix, θFi∗ is the optimal parameter vector, and εFix is the approximate error. Defining θ˜=θ∗−θ^, one has(8)fix=li−1Fix=li−1θFi∗TφFix+li−1εFix=li−1θ˜FiT+θ˜FiTφFix+li−1εFix=li−1θ˜FiTφFix+li−1θ^FiTφFix+li−1εFix.
Let us denoteL−1θ˜FTφFx=l1−1θ˜F1TφF1x,l2−1θ˜F2TφF2x,…,ln−1θ˜FnTφFnxT, L−1θ^FTφFx=l1−1θ^F1TφF1x,l2−1θ^F2TφF2x,…,ln−1θ^FnTφFnxT, and L−1εFx=l1−1εF1x,l2−1εF2x,…,ln−1εFnxT. So, f(x) in (6) can be expressed as(9)fx=L−1θ˜FTφFx+L−1θ^FTφFx+L−1εFx.
LetD=L−1εFx+d¯t. The error system of e can be described as(10)e˙=L−1θ˜FTφFx+mGxu+L−1θ^FTφFx+D−x˙d.Remark 3.
From (10), one knows that the unknown vector D is composed by the external disturbance d(t), the approximate error ɛF, and Δu. So, the disturbance observer D^ will be constructed so that D^+θFTφFx can estimate d(t) + f(x) + Δu in this paper.
The aim of this paper is (i) to use a given performance function to get rid of the dependence on the sign of the initial valuee(0); (ii) to design the composite learning control with the disturbance observer to make sure that the tracking error e satisfies the prescribed performance boundary; and (iii) to estimate system uncertainties accurately.


## 3. Control Design and Stability Analysis

### 3.1. Prescribed Performance

It is assumed here that the tracking errorei satisfies the following prescribed performance boundary (PPB) for i = 1, 2, …, n:(11)−λ¯yit<ei<λ¯yit,where λ¯andλ¯ are two adjustable positive constants and yi(t) is a performance function. Notice that xi(0) is unknown, which leads to ei(0) is unknown. To make sure that −λ¯yi0<ei0<λ¯yi0 holds, one chooses the performance function yi(t) [17] as(12)yit=cothλ1it+λ2i−1+λ∞,where λ1i, λ2i, and λ∞ are designed positive constants. Obviously, yi(t) has the following properties:(1)
yi(t) is positive and decreasing(2)
yi0=cothλ2i−1+λ∞=e2λ2i+1/e2λ2i−1−1+λ∞>λ∞;(3)
limλ2i⟶0+yi0=+∞(4)
limt⟶+∞yit=λ∞By using property 3 of the performance functionyi(t), we know that −λ¯limλ2i⟶0+yi0=−∞ and λ¯limλ2i⟶0+yi0=+∞. Then, for any unknown bounded initial value ei(0), we can select the value of λ2i adequately small so that the following inequality holds:(13)−λ¯yi0<ei0<λ¯yi0.Next, an error transformation functionS(zi) will be introduced to transform ei into a transformation error signal zi. Let(14)ei=yitSzi,where Szi=λ¯ezi−λ¯e−zi/ezi+e−zi. Apparently, S(zi) has the following properties:(1)
S(zi) is a strictly increasing smooth function(2)
limzi⟶−∞Szi=−λ¯,limzi⟶+∞Szi=λ¯From properties 1 and 2 ofS(zi), one knows that −λ¯<Szi<λ¯. Multiplying both sides of the above inequality by yi(t), we have −λ¯yit<yitSzi<λ¯yit. Therefore, one can use (14) to represent PPB (11).Now,yi is denoted as yi(t). Because S(zi) is strictly increasing function, one obtains(15)zi=S−1zi=Teiyi=12lnei/yi+λ¯λ¯−ei/yi,and the following transformation error system(16)z˙i=rie˙i−y˙iyiei,where ri=1/2yi1/ei/yi+λ¯−1/ei/yi−λ¯>0 and y˙it=λ1i−λ1icothλ1it+λ2i2.Lemma 1.
Ifzi is bounded, then PPB (11) holds.Proof.
Sincezi is bounded, there exists a positive constant M such that zi≤M. Noting that Si(zi) is strictly monotonic increasing, one has −λ¯=limM⟶+∞S−M<S−M≤Szi=ei/yi≤SM<limM⟶+∞SM=λ¯, that is, −λ¯yi<ei<λ¯yi holds.
Now, according to (10) and (11), the transformation system is obtained as follows:(17)z˙=rL−1θ^FTφFx+mGxu+L−1θ˜FTφFx+D−x˙d−y˙ey,where r = diag(r1, r2, …, rn) and y˙e/y=y˙1e1/y1,y˙2e2/y2,…,y˙nen/ynT. In order to show the performance of FLSs, z^ is introduced as the estimation of z, and(18)z^˙=r−Kz˜+L−1θ^FTφFx+mGxu+D^−x˙d−y˙ey,where z˜=z^−z and K = diag(k1, k2, …, kn), ki is the positive design constant.
The disturbance observerD^ is constructed as(19)D^=l0z−v,v˙=rL−1θ^FTφFx+mGxu+D^−x˙d−y˙ey+l0−1−z+Πz˜,where Π = diag(π1, π2, …, πn), πi and l0 are positive design parameters.
The derivative ofD^ yields(20)D^˙=l0z˙−v˙=l0rL−1θ˜FTφFx+D˜+rz−rΠz˜,where D˜=D−D^. Furthermore, one obtains(21)D˜˙=D˙−rz+rΠz˜−l0rL−1θ˜FTφFx+D˜.


### 3.2. Main Result

Based on the above analysis, the main result of this paper is given.Theorem 1.
For nonlinear system (1) with unknown initial value x(0) and the time-varying external disturbance d(t), if the controller u(t) and the composite learning-based parameter adaptive laws are designed as(22)u=1mG−1x−Cz−L−1θ^FφFx−D^+ye.y+x˙d,θ^˙Fi=ηFili−1ziφFix−πili−1z˜iφFix−δiθ^Fi,where C = diag(c1, c2, …, cn), ci,δi,andηFi are positive design constants, i = 1, 2, …, n, and the disturbance observer is designed as (19), then all the close-loop system signals in (23) are uniformly ultimately bounded. Furthermore, the error state ei satisfies PPB (11).Proof.
Define the Lyapunov function as(23)V=V1+V2+V3+V4,where(24)V1=12zTr−1z=∑i=1n12zi2ri,V2=12D˜Tr−1D˜=∑i=1n12D˜i2ri,V3=∑i=1n12ηFiθ˜FiTθ˜Fi,V4=12z˜Tr−1Πz˜=∑i=1n12πiz˜i2ri.
Then, the derivatives ofVi, i = 1, 2, 3, 4 can be given as follows:(25)V˙1=∑i=1nziz˙iri−r˙izi22ri2=∑i=1nzili−1θ^FiTφFix+mgix−y˙ieiyi−x˙di+Dit+li−1θ˜FiTφFix−r˙izi22ri2=∑i=1n−ci+r˙i2ri2zi2+ziD˜i+li−1ziθ˜FiTφFix,(26)V˙2=∑i=1nD˜iD˜˙iri−r˙iD˜i22ri2=∑i=1n−l0+r˙i2ri2D˜i2−D˜izi+πiz˜iD˜i+D˜iD˙iri−l0li−1D˜iθ˜FiTφFix,(27)V˙3=∑i=1n1ηFiθ˜FiTθ˜˙Fi=∑i=1n−li−1ziθ˜FiTφFix+πili−1z˜iθ˜FiTφFix+δiθ˜FiTθ^Fi,(28)V˙4=∑i=1nπiziz˜˙iri−πir˙iz˜i22ri2=∑i=1nπiz˜i−kiz˜i−D˜i−li−1θ˜FiTφFix+πir˙iz˜i22ri2=∑i=1n−kiπi+πir˙i2ri2z˜i2−πiz˜iD˜i−πili−1z˜iθ˜FiTφFix.
Using (25)–(28), one has(29)V˙=∑i=1n−ci+r˙i2ri2zi2−l0+r˙i2ri2D˜i2−kiπi+πir˙i2ri2z˜i2−l0li−1D˜iθ˜FiTφFix+D˜iD˙iri+δiθ˜FiTθ^Fi.
According to Meng and Moore [20], there exist unknown positive constants χi and ϱi such that φFix≤χi and ‖D˙i‖ ≤ φi, respectively. So, the following inequalities hold:(30)D˜iD˙iri≤D˜i22ri2+D˙i22≤D˜i22ri2+ϱi22,D˜iθ˜FiTφFix≤νiχi2D˜i22+θ˜Fi22νi,δiθ˜FiTθ^Fi≤−δi2θ˜2+δi2θ∗2,where νi is a positive constant. Substituting (30) into (29), one can obtain(31)V˙=∑i=1n−ci+r˙i2ri2zi2−l0+r˙i2ri2−νil0l−1χi22D˜i2−kiπi+πir˙i2ri2z˜i2−δi2−l0l−12νiθ˜Fi2︸θ˜FiTθ˜Fiθ˜Fi+R0,where R0=∑i=1nϱi2/2+δi/2θFi∗s2. By choosing parameters l0, li, and δi to satisfy ci>−r˙i/2ri2, l0>−r˙i/2ri2+νil0l−1χi2/2, ki>−r˙i/2ri2, and δi>l0l−1/νi, and define the following compact sets(32)Ωzi=zizi≤R0ci+r˙i/2ri2,ΩD˜i=D˜iD˜i≤R0l0+r˙i/2ri2−νil0l−1χi2/2,Ωθi=θ˜iθ˜i≤R0δi/2−l0l−1/2νi,Ωz˜i=z˜iz˜i≤R0kiπi+πir˙i/2ri2.
Obviously, ifzi∉Ωzi or D˜i∉ΩD˜i or θ˜i∉Ωθi or z˜i∉Ωz˜i, V˙ will be negative. Hence, zi,D˜i,θ˜i, and z˜i are uniformly bounded. Because zi is bounded, it is known from Lemma 1 that ei satisfies PPB (11).Remark 4.
In (22), a composite learning adaptation law for the fuzzy parameter θ^ is given. It should be noticed that this law is very important to achieve accurate approximation of unknown nonlinear functions in the PPC design. In fact, adaptive fuzzy PPC methods are used in many references, for example, in [27–30]. However, the control methods in the aforementioned literature have no such ability.Remark 5.
In the proposed method, as indicated in Theorem1, all signals will remain bounded. It has also been shown that the variables zi,D˜i,θ˜i,andz˜i will converge to a small region of zero whose radius is determined by control design parameters. One can adjust the radius of the region by using different parameters.


## 3.1. Prescribed Performance

It is assumed here that the tracking errorei satisfies the following prescribed performance boundary (PPB) for i = 1, 2, …, n:(11)−λ¯yit<ei<λ¯yit,where λ¯andλ¯ are two adjustable positive constants and yi(t) is a performance function. Notice that xi(0) is unknown, which leads to ei(0) is unknown. To make sure that −λ¯yi0<ei0<λ¯yi0 holds, one chooses the performance function yi(t) [17] as(12)yit=cothλ1it+λ2i−1+λ∞,where λ1i, λ2i, and λ∞ are designed positive constants. Obviously, yi(t) has the following properties:(1)
yi(t) is positive and decreasing(2)
yi0=cothλ2i−1+λ∞=e2λ2i+1/e2λ2i−1−1+λ∞>λ∞;(3)
limλ2i⟶0+yi0=+∞(4)
limt⟶+∞yit=λ∞By using property 3 of the performance functionyi(t), we know that −λ¯limλ2i⟶0+yi0=−∞ and λ¯limλ2i⟶0+yi0=+∞. Then, for any unknown bounded initial value ei(0), we can select the value of λ2i adequately small so that the following inequality holds:(13)−λ¯yi0<ei0<λ¯yi0.Next, an error transformation functionS(zi) will be introduced to transform ei into a transformation error signal zi. Let(14)ei=yitSzi,where Szi=λ¯ezi−λ¯e−zi/ezi+e−zi. Apparently, S(zi) has the following properties:(1)
S(zi) is a strictly increasing smooth function(2)
limzi⟶−∞Szi=−λ¯,limzi⟶+∞Szi=λ¯From properties 1 and 2 ofS(zi), one knows that −λ¯<Szi<λ¯. Multiplying both sides of the above inequality by yi(t), we have −λ¯yit<yitSzi<λ¯yit. Therefore, one can use (14) to represent PPB (11).Now,yi is denoted as yi(t). Because S(zi) is strictly increasing function, one obtains(15)zi=S−1zi=Teiyi=12lnei/yi+λ¯λ¯−ei/yi,and the following transformation error system(16)z˙i=rie˙i−y˙iyiei,where ri=1/2yi1/ei/yi+λ¯−1/ei/yi−λ¯>0 and y˙it=λ1i−λ1icothλ1it+λ2i2.Lemma 1.
Ifzi is bounded, then PPB (11) holds.Proof.
Sincezi is bounded, there exists a positive constant M such that zi≤M. Noting that Si(zi) is strictly monotonic increasing, one has −λ¯=limM⟶+∞S−M<S−M≤Szi=ei/yi≤SM<limM⟶+∞SM=λ¯, that is, −λ¯yi<ei<λ¯yi holds.
Now, according to (10) and (11), the transformation system is obtained as follows:(17)z˙=rL−1θ^FTφFx+mGxu+L−1θ˜FTφFx+D−x˙d−y˙ey,where r = diag(r1, r2, …, rn) and y˙e/y=y˙1e1/y1,y˙2e2/y2,…,y˙nen/ynT. In order to show the performance of FLSs, z^ is introduced as the estimation of z, and(18)z^˙=r−Kz˜+L−1θ^FTφFx+mGxu+D^−x˙d−y˙ey,where z˜=z^−z and K = diag(k1, k2, …, kn), ki is the positive design constant.
The disturbance observerD^ is constructed as(19)D^=l0z−v,v˙=rL−1θ^FTφFx+mGxu+D^−x˙d−y˙ey+l0−1−z+Πz˜,where Π = diag(π1, π2, …, πn), πi and l0 are positive design parameters.
The derivative ofD^ yields(20)D^˙=l0z˙−v˙=l0rL−1θ˜FTφFx+D˜+rz−rΠz˜,where D˜=D−D^. Furthermore, one obtains(21)D˜˙=D˙−rz+rΠz˜−l0rL−1θ˜FTφFx+D˜.


## 3.2. Main Result

Based on the above analysis, the main result of this paper is given.Theorem 1.
For nonlinear system (1) with unknown initial value x(0) and the time-varying external disturbance d(t), if the controller u(t) and the composite learning-based parameter adaptive laws are designed as(22)u=1mG−1x−Cz−L−1θ^FφFx−D^+ye.y+x˙d,θ^˙Fi=ηFili−1ziφFix−πili−1z˜iφFix−δiθ^Fi,where C = diag(c1, c2, …, cn), ci,δi,andηFi are positive design constants, i = 1, 2, …, n, and the disturbance observer is designed as (19), then all the close-loop system signals in (23) are uniformly ultimately bounded. Furthermore, the error state ei satisfies PPB (11).Proof.
Define the Lyapunov function as(23)V=V1+V2+V3+V4,where(24)V1=12zTr−1z=∑i=1n12zi2ri,V2=12D˜Tr−1D˜=∑i=1n12D˜i2ri,V3=∑i=1n12ηFiθ˜FiTθ˜Fi,V4=12z˜Tr−1Πz˜=∑i=1n12πiz˜i2ri.
Then, the derivatives ofVi, i = 1, 2, 3, 4 can be given as follows:(25)V˙1=∑i=1nziz˙iri−r˙izi22ri2=∑i=1nzili−1θ^FiTφFix+mgix−y˙ieiyi−x˙di+Dit+li−1θ˜FiTφFix−r˙izi22ri2=∑i=1n−ci+r˙i2ri2zi2+ziD˜i+li−1ziθ˜FiTφFix,(26)V˙2=∑i=1nD˜iD˜˙iri−r˙iD˜i22ri2=∑i=1n−l0+r˙i2ri2D˜i2−D˜izi+πiz˜iD˜i+D˜iD˙iri−l0li−1D˜iθ˜FiTφFix,(27)V˙3=∑i=1n1ηFiθ˜FiTθ˜˙Fi=∑i=1n−li−1ziθ˜FiTφFix+πili−1z˜iθ˜FiTφFix+δiθ˜FiTθ^Fi,(28)V˙4=∑i=1nπiziz˜˙iri−πir˙iz˜i22ri2=∑i=1nπiz˜i−kiz˜i−D˜i−li−1θ˜FiTφFix+πir˙iz˜i22ri2=∑i=1n−kiπi+πir˙i2ri2z˜i2−πiz˜iD˜i−πili−1z˜iθ˜FiTφFix.
Using (25)–(28), one has(29)V˙=∑i=1n−ci+r˙i2ri2zi2−l0+r˙i2ri2D˜i2−kiπi+πir˙i2ri2z˜i2−l0li−1D˜iθ˜FiTφFix+D˜iD˙iri+δiθ˜FiTθ^Fi.
According to Meng and Moore [20], there exist unknown positive constants χi and ϱi such that φFix≤χi and ‖D˙i‖ ≤ φi, respectively. So, the following inequalities hold:(30)D˜iD˙iri≤D˜i22ri2+D˙i22≤D˜i22ri2+ϱi22,D˜iθ˜FiTφFix≤νiχi2D˜i22+θ˜Fi22νi,δiθ˜FiTθ^Fi≤−δi2θ˜2+δi2θ∗2,where νi is a positive constant. Substituting (30) into (29), one can obtain(31)V˙=∑i=1n−ci+r˙i2ri2zi2−l0+r˙i2ri2−νil0l−1χi22D˜i2−kiπi+πir˙i2ri2z˜i2−δi2−l0l−12νiθ˜Fi2︸θ˜FiTθ˜Fiθ˜Fi+R0,where R0=∑i=1nϱi2/2+δi/2θFi∗s2. By choosing parameters l0, li, and δi to satisfy ci>−r˙i/2ri2, l0>−r˙i/2ri2+νil0l−1χi2/2, ki>−r˙i/2ri2, and δi>l0l−1/νi, and define the following compact sets(32)Ωzi=zizi≤R0ci+r˙i/2ri2,ΩD˜i=D˜iD˜i≤R0l0+r˙i/2ri2−νil0l−1χi2/2,Ωθi=θ˜iθ˜i≤R0δi/2−l0l−1/2νi,Ωz˜i=z˜iz˜i≤R0kiπi+πir˙i/2ri2.
Obviously, ifzi∉Ωzi or D˜i∉ΩD˜i or θ˜i∉Ωθi or z˜i∉Ωz˜i, V˙ will be negative. Hence, zi,D˜i,θ˜i, and z˜i are uniformly bounded. Because zi is bounded, it is known from Lemma 1 that ei satisfies PPB (11).Remark 4.
In (22), a composite learning adaptation law for the fuzzy parameter θ^ is given. It should be noticed that this law is very important to achieve accurate approximation of unknown nonlinear functions in the PPC design. In fact, adaptive fuzzy PPC methods are used in many references, for example, in [27–30]. However, the control methods in the aforementioned literature have no such ability.Remark 5.
In the proposed method, as indicated in Theorem1, all signals will remain bounded. It has also been shown that the variables zi,D˜i,θ˜i,andz˜i will converge to a small region of zero whose radius is determined by control design parameters. One can adjust the radius of the region by using different parameters.


## 4. Example

In this section, Liu chaotic system [31] is introduced to show the effect of proposed control method (22), and the Liu chaotic system is described as follows:(33)x˙1=−x1−x22︸f1x+d1t+g1xΓ1u1,x˙2=2.5x2−4x1x3︸f2x+d2t+g2xΓ2u2,x˙3=−5x3+4x1x2︸f3x+d3t+g3xΓ3u3,where g1x = g2x = g3x = 1, d1(t) = 5.5 sin(t), d2(t) = 6.0 sin(t), and d3(t) = 6.5 sin(t). The reference signal xd = [cos(t), 0.5 cos(t), −2.6 cos(t)]T. The state initial value is assumed unknown. In this case, the traditional PPC method cannot be applied. Assume that the tracking error is in the range [−5, 5], select the performance function y1(t) = y2(t) = y3(t) = coth(0.5t + 0.25) − 1 + 0.1, and initial values are chosen as x(0) = [−1, −0.5, −0.8]T, z^0=1.2,1.2,1.2T, v(0) = [−2, 2, 3]T, and θ^Fi0=0, and parameters are chosen as m=3,bri=bli=4,λ¯=λ¯=1, l0 = 20, li = 10, ki = ci = 10, and ηFi=πi=3,δi=0.5, i = 1, 2, 3. The fuzzy membership functions are selected as(34)φfk=exp−12k+7.5−2.5j1.22,where j = 1, 2, 3, 4, 5; k = x1, x2, x3.The simulation results are shown in Figures1–6. Denote the proposed method of this paper as OBPPC. From Figures 1–3, one can see that tracking errors e1, e2, and e3 are limited within PPB (11). Under the disturbance observer (19), D^1+l1−1θ^F1TφF1x, D^2+l2−1θ^F2TφF2x, and D^3+l3−1θ^F3TφF3x can efficiently estimate f1(x) + d1(t) + Δu1, f2(x) + d2(t) + Δu2, and f3(x) + d3(t) + Δu3 in Figure 4. Here, the existence problem of ṙi in (32) needs to be considered. In Figure 6, the tracking errors ei enter the stable region [−0.1, 0.1], but the dead-zone fluctuation of ei will occur due to the influence of the dead-zone inputs ui (see Figure 5), which will cause ri to fluctuate. Notice that its derivative r˙i is bounded, and the value of r˙i/2ri2 is not larger. Therefore, this paper chooses ci = 30 which can guarantee the existence of compact sets Ωzi,ΩD˜i, and Ωz˜i.Figure 1
The trajectories of (a)xd1 and x1 and (b) e1.
(a)
(b)Figure 2
The trajectories of (a)xd2 and x2 and (b) e2.
(a)
(b)Figure 3
The trajectories of (a)xd3 and x3 and (b) e3.
(a)
(b)Figure 4
Estimation of (a)f1(x) + d1(t) + Δu1, (b) f2(x) + d2(t) + Δu2, and (c)f3(x) + d3(t) + Δu3 by using OBPPC.
(a)
(b)
(c)Figure 5
The trajectories ofu1, u2, and u3 by using OBPPC.Figure 6
The trajectories of (a) tracking errorei, (b) ri, (c) ṙi, and (d)r˙i/2ri2 (i = 1, 2, 3) by using OBPPC.
(a)
(b)
(c)
(d)In order to compare with the traditional PPC, it is necessary to assume that the sign of the tracking errorei is known, and the control method law is designed as(35)ui=−ϑ^f¯iTφf¯ix−30z¯i+eiμ˙iμi+xdi,ϑ^˙f¯i=40z¯1φf¯ix−0.3ϑ^f¯1,z¯i=tanheiμi,μi=5.0−0.1exp−5.0t+0.1,where f¯ix=fix+dit+Δui and ϑ^f¯iTφf¯ix is the estimation of f¯ix. The parameter λ in (1) is selected as λ = 1. Denote the control law (35) as TPPC. It can be seen from Figure 7 that the error ei satisfies (1), but the estimation of f¯ix is poor. Therefore, it can be seen from the comparison that the method proposed in this paper is more effective.Figure 7
The trajectories of (a) tracking errore1, e2, e3 and estimation of (b) f1(x) + d1(t) + Δu1, (c) f2(x) + d2(t) + Δu2, and (d) f3(x) + d3(t) + Δu3 by using TPPC.
(a)
(b)
(c)
(d)


## 5. Conclusion

By using the composite PPC method and the disturbance observer, this paper addresses the stability for a class of uncertain nonlinear system with unknown initial states. Firstly, a performance function is introduced to overcome the problem of unknown initial values. Then, the error system is transformed to an equivalent one by using transformation function. Through the constructed observer and fuzzy logic systems, unknown functions and time-varying external disturbances can be estimated effectively. Meanwhile, the boundedness of all involved signals is guaranteed. Moreover, the tracking errors satisfy the prescribed performance bounds all the time. How to apply the proposed method to real-world systems, such as robotic manipulators, is one of our future research directions.



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*Source: 2948130-2020-03-26.xml*