# An Integral Sliding Mode Control of Uncertain Chaotic Systems via Disturbance Observer

Authors: Hua Zhang
Journal: Complexity
 (2021)

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## Abstract

This paper proposes an integral sliding mode control (ISMC) method of a class of uncertain chaotic systems with saturation inputs. Firstly, fuzzy logic system (FLS) is used to estimate the unknown nonlinear function. Then, a disturbance observer is constructed to estimate a compound disturbance, which contains the external disturbance, the error of saturation input and control output, and the fuzzy estimation error. Subsequently, a proposed integral sliding mode controller can ensure that all signals of the closed-loop system are ultimately bounded, and based on the dynamic system of the integral sliding mode variable itself, the ultimate bound of the tracking error can be estimated. Simulation results show that the proposed ISMC method is more effective than the traditional ISMC method.


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## Body

## 1. Introduction

Due to the unpredictability and the sensitivity to initial conditions, the chaotic system makes itself useful in many fields such as information processing, secure communications, and mechanical systems [1, 2]. However, some actual systems, such as flutter of aircraft wings and vibration of horizontal platforms, may cause bad results if the chaotic behavior occurs. Therefore, it is necessary to suppress such harmful chaotic behavior. At present, many methods have been proposed to stabilize or synchronize chaotic systems, such as adaptive control method [3, 4], backstepping control [5, 6], impulsive control [7], intermittent control [8], and sliding mode control [9–14].As we all know, the advantage of using sliding mode control method to control chaotic systems and other nonlinear systems is its good robustness. For example, Wang et al. [11] proposed a nonsingular terminal SMC method to ensure that all states of the chaotic system reach the designed sliding surface in finite time. By using adaptive terminal SMC, Yang and Ou [12] studied the synchronization problem of chaotic gyros. Similarly, by employing adaptive SMC, Chen et al. [13] investigated the synchronization for multiple uncertain coupled chaotic systems. It can be seen that the abovementioned sliding mode control method only considers whether the arrival condition of the sliding surface can be satisfied and does not consider the estimation problem for system uncertainties. Meanwhile, the hysteresis of control switching will cause chattering phenomenon. In order to eliminate this disadvantage, Haghighi and Mobayen [15] proposed a high-order terminal SMC technique for a class of fourth-order systems. However, the higher-order sliding mode controller includes larger amount of higher-order derivatives of sliding mode variable, which may lead to increased noise in closed-loop systems.Recently, in order to estimate system uncertainties or external disturbances, Zhou et al. [16–20] explored the construction principle of disturbance observer and combined different control methods to achieve the stability of closed-loop systems. Based on the disturbance observer and fuzzy terminal SMC, Vahidi-Moghaddam et al. [17] investigated the finite-time asymptotic stability of uncertain MIMO systems. Xu in [18] proposed composite terminal SMC learning control schemes for quadrotor dynamics via the disturbance observer. However, control results of [17–20] only ensure that all signals of the closed-loop system approach a small region, but the ultimate bound of this area has not been estimated.Inspired by the abovementioned works, this paper investigates the problem of integral sliding mode control for a class of uncertain chaotic systems with saturation inputs, and the proposed integral SMC method can not only guarantee all signals of the closed-loop system are bounded but also accurately estimate system uncertainties. The main contributions of this paper are summarized as follows:(1)
Compared with the traditional integral sliding mode variable, the integrated sliding mode variable proposed in this paper can estimate the error bound by its own dynamics when the boundedness of the integrated sliding mode variable is guaranteed.(2)
The established disturbance observer and fuzzy parameter law can accurately estimate system uncertainties.(3)
Compared with the traditional ISMC method, the proposed method in this paper can eliminate the chattering phenomenon.The rest of this work is arranged as follows. Some assumptions and lemmas and the problem statement are presented in Section2. Section 3 gives sliding mode control design and stability analysis. In Section 4, some comparison results are presented to show the validity of the proposed method. At last, the conclusion is included in Section 5.


## 2. Preliminaries

In general, the chaotic system can be described as(1)x.=Fx,where x=x1,x2,…,xnT∈Rn is the state vector and Fx=f1x,f2x,…,fnxT is the nonlinear vector function. Many classic chaotic systems such as Lorenz chaotic system, Chen chaotic system, and Lu.. chaotic system can be abbreviated as the form of system (1), and the controlled chaotic system (1) is expressed as(2)x.=Fx+dt+satu,where dt=d1t,d2t,…,dntT is the external disturbance vector, u=u1,u2,…,unT is the control output vector, and satu=satu1,satu2,…,satunT is the control input vector subject to saturation type nonlinearity, and satui is described as(3)satui=ui∗·signui,ifui≥ui∗,ui,ifui<ui∗,where ui∗ is the unknown bound of symmetric input saturation.Define the tracking error vectore=x−xd=e1,e2,…,enT, and xd=xd1,xd2,…,xdnT is the reference signal. According to (2), we obtain the following error dynamic system:(4)e.=Fx+dt+satu−x.d.In order to achieve effective tracking of the reference signalxd by the state x, the following assumptions about Fx, xd, and dt are made.Assumption 1.
The nonlinear functionFx is unknown but bounded, and the state x is measurable.Assumption 2.
The reference signalxd is C2, and xd and xd. are available.Assumption 3.
The time-varying external disturbancedt and its derivative are unknown but bounded.DefineΔu=Δu1,Δu2,…,ΔunT∈Rn, where Δui=satui−ui, i=1,2,…,n.Remark 1.
For the error system (4), we can use method [21] to track the reference signal xd. Firstly, let Fx=F1x¯+ΔF, where Fx¯ is known. Define Dt=dt+Δu+ΔF, and assume that D.≤δ, where δ is unknown. Then, construct the disturbance observer D^ as D^=η^+K1e, where η^ is the intermediate variable and satisfies(5)η^.=−K1F1x¯+u+η^+K1e,where K1=diagk11,…,k1n,k1j is the positive constant, and design the following control law u as(6)u=−F1x¯−D^−K2e,where K2=diagk21,…,k2n, k2j,j=1,2,…,n are the positive constants. This method can guarantee that e and D˜ are uniformly bounded. However, this method does not cover two situations: one is that the function Fx is unknown; the other is whether Fx+D can be accurately estimated. In order to consider the abovementioned two situations, this paper will use the integral sliding mode control method combined with the disturbance observer and fuzzy logic technology to design a control method to realize the fast stability of tracking error et. In addition, the controller and error state will effectively eliminate chattering phenomenon.Definef¯ix=lifix, where li is the positive design constant. By using FLSs, f¯ix can approximate as(7)f¯ix=θf¯i∗Tφf¯ix+εf¯i,εf¯i≤εiM,where θf¯i∗∈Rmi are the ideal weight vectors, εf¯i is the approximation error, and εiM is the upper bound of εf¯i. φf¯ix=φf¯i1x,φf¯i2x,…,φf¯imixT∈Rmi, where φf¯ijx, j=1,…,mi are the Gaussian functions. Obviously, there exists a positive constant p∗ such that φf¯ix≤p∗.Defineθ˜f¯i=θf¯i∗−θ^f¯i, where θ^f¯i is the estimation of θf¯i∗. So, we obtain(8)fix=li−1f¯ix=li−1θ˜f¯iTφf¯ix+li−1θ^f¯iTφf¯ix+li−1εf¯ix.DenoteL−1θ˜FTφFx=l1−1θ˜f¯1Tφf¯1x,l2−1θ˜f¯2Tφf¯2x,…,ln−1θ˜f¯nTφf¯nxT, L−1θ^FTφFx=l1−1θ^f¯1Tφf¯1x,l2−1θ^f¯2Tφf¯2x,…,ln−1θ^f¯nTφf¯nxT, L−1εFx=l1−1εf¯1x,l2−1εf¯2x,…,ln−1εf¯nxT. So, Fx in (4) can be expressed as(9)Fx=L−1θ˜FTφFx+L−1θ^FTφFx+L−1εFx.LetD=D1,D2,…,DnT=L−1εFx+dt+Δu. The error system (4) can be rewritten as(10)e.=L−1θ˜FTφFx+L−1θ^FTφFx+u+D−x.d.Because the compounded disturbanceD is unknown, it cannot appear directly in the controller design. But from a practical system point of view, D is clearly bounded. Combining with the structural ideas of disturbance observer [18, 21, 22], we also make the following assumption.Assumption 4.
The derivative ofD satisfies D.≤D∗, where D∗ is an unknown positive constant.The following lemmas are introduced for the subsequent discussions.Lemma 1 (see [23]).
Let0<p1<p2, then(11)∑i=1naip21/p2≤∑i=1naip11/p1,where ai is the real number.Lemma 2 (see [24]).
For anyz∈R, 1>γ≥0, and ϵ>0, the following inequality holds:(12)zγ+1−zzγtanhzϵ≤ηγϵγ+1,where ηγ=ϱγ2ϱ−γ−10 and ϱ is the unique solution of ϱ1+tanhϱ=γ+1.Remark 2.
Whenγ=0, we get a famous inequality: z−ztanhz/ϵ≤η0ϵ, where η0≈0.2758. In this paper, since both the sliding mode and the controller contain tanh· functions, the inequalities (12) of Lemma 2 play an important role.The aim of this paper is (1) to propose an effective control method so that the error statee can be remained within a small neighborhood of zero; (2) to construct the disturbance observer so that Fx+dt+Δu can be accurately estimated; (3) to design a new integral sliding variable such that the chatter phenomenon of controller can be eliminated.


## 3. Sliding Mode Control Design and Stability Analysis

### 3.1. Construction of Integral Sliding Mode and Disturbance Observer

In order to design a sliding mode robust controller to make the tracking errore stable, the integral sliding variable needs to be designed. In [25], an integral terminal sliding variable is designed as(13)s=e+∫0tμ1Signγ1edτ,where s=s1,s2,…,snT∈Rn, Signγ1e=e1γ1signe1,e2γ1signe2,…,enγ1signenT, γ1∈0,1, and μ1 is a positive parameter. By using the traditional sliding mode control method, the occurrence of chatter phenomenon is mainly due to the hysteresis of control switching. To avoid the chatter phenomenon, we modify the sliding surface (13) as follows:(14)s=e+∫0tμ1Tanhϵ1γ1e+μ2Tanhϵ10edτ,where Tanhϵ1γ1e=e1γ1tanhe1/ϵ1,e2γ1tanhe2/ϵ1,…,enγ1tanhen/ϵ1T, Tanhϵ10e=tanhe1/ϵ1,tanhe2/ϵ1,…,tanhen/ϵ1T, μ2 is the designed positive parameter and ϵ1 is a small positive constant, and γ1 and μ1 are the same as in (13).Remark 3.
In general, the aim of using the integral sliding variables in (13) is to improve its control performance and eliminate the reaching phase under the nominal control law. Different from the former, based on the dynamic system of integral sliding variable s in (14), this paper is to explore the ultimate boundary of the tracking error e.According to (10) and (14), the dynamic system of s is obtained as follows:(15)s.=L−1θ˜FTφFx+L−1θ^FTφFx+u+D−x.d+μ1Tanhϵ1γ1e+μ2Tanhϵ10e.In order to show the performance of RBFNNs, definings^=s^1,s^2,…,s^nT as the estimation of s, we have(16)s^.=−μ3s˜+L−1θ^FTφFx+u+D^−x.d+μ1Tanhϵ1γ1e+μ2Tanhϵ10e,where s˜=s^−s, μ3 is the positive design constant, and D^=D^1,D^2,…,D^nT is the estimation of D. Furthermore, the derivative of s˜ is obtained as(17)s˜.=−μ3s˜−L−1θ˜FTφFx−D˜.Now, the disturbance observerD^ is constructed as(18)D^=μ4s−w,w.=L−1θ^FTφFx+u+D^−x.d+μ1Tanhϵ1γ1e+μ2Tanhϵ10e+μ4−1−s+μ5s˜,where w=w1,w2,…,wnT∈Rn is an auxiliary variable vector and μ4 and μ5 are positive design parameters.The derivative ofD^ can be written as(19)D^.=μ4s.−w.=μ4L−1θ˜FTφFx+D˜+s−μ5s˜,where D˜=D−D^=D˜1,D˜2,…,D˜nT, and then, we obtain(20)D˜.=D.−s+μ5s˜−μ4L−1θ˜FTφFx+D˜.


### 3.2. Main Results

Now, the controllerut is designed as(21)u=−μ6s−μ7Tanhϵ2γ2s−L−1θ^FTφFx−D^+x.d−μ1Tanhϵ1γ1e−μ2Tanhϵ10e,and the parameter adaptive laws are proposed as(22)θ^.f¯i=ηf¯ili−1siφf¯ix−μ5li−1s˜iφf¯ix−δ1θ^f¯i,where Tanhϵ2γ2s=s1γ2tanhs1/ϵ2,s2γ2tanhs2/ϵ2,…,snγ2tanhsn/ϵ2T, δ1,μ6,μ7, and ηf¯i are the design parameters, i=1,2,…,n, and ϵ2 is a small positive constant. We give the main result as follows.Theorem 1.
For the error system (10) under Assumptions 1–4 and the disturbance observer (18), the controller (21) and parameter adaptive laws (22) guarantee that all the closed-loop system signals in (23) are uniformly ultimately bounded.Proof.
Consider the Lyapunov function as(23)V=V1+V2+V3+V4,where(24)V1=12sTs=12s2=∑i=1n12si2,V2=12D˜TD˜=12D˜2=∑i=1n12D˜i2,V3=μ52s˜Ts˜=μ52s˜2=∑i=1nμ52s˜i2,V4=∑i=1n12ηf¯iθ˜f¯i2=∑i=1n12ηf¯iθ˜f¯iTθ˜f¯i.
Substituting (21) into (15), the derivative of V1 can be obtained as(25)V˙1=sTs.=−μ6sTs−μ7sTTanhϵ2γ2s+sTD˜+sTL−1θ˜FTφFx=−μ6sTs−∑i=1nμ7sisiγ2tanhsiϵ2+sTD˜+∑i=1nsili−1θ˜f¯iTφf¯x.
From (20), the derivative of V2 yields(26)V˙2=D˜TD˜.=D˜TD.−μ4D˜TD˜−D˜Ts+μ5D˜Ts˜−∑i=1nμ4D˜ili−1θ˜f¯iTφf¯ix.
From (17), the derivative of V3 yields(27)V˙3=μ5s˜Ts˜.=−μ3μ5s˜Ts˜−μ5s˜TD˜−∑i=1nμ5li−1s˜iθ˜f¯iTφf¯ix.
From (22), the derivative of V4 yields(28)V˙4=∑i=1n1ηf¯iθ˜f¯iTθ˜.f¯i=∑i=1n−li−1siθ˜f¯iTφf¯ix+μ5li−1s˜iθ˜f¯iTφf¯ix+δ1θ˜f¯iTθ^f¯i.
Using (25), (26), (27) and (28), we have(29)V˙=−μ6sTs−∑i=1nμ7sisiγ2tanhsiϵ2+D˜TD.−μ4D˜TD˜−∑i=1nμ4D˜ili−1θ˜f¯iTφf¯ix−μ3μ5s˜Ts˜+∑i=1nδ1θ˜f¯iTθ^f¯i.
Since the following inequalities hold,(30)D˜TD.≤D˜22+D.22≤D˜22+D∗22,−μ4li−1D˜iθ˜f¯iTφf¯ix≤μ4p∗2D˜i22lmin+μ4θ˜f¯i22lmin,δ1θ˜f¯iTθ^f¯i≤−δ12θ˜f¯i2+δ12θf¯i∗2,where lmin=minl1,l2,…,ln.
By using Lemma1, one has(31)−∑i=1nμ7sisiγ2tanhsiϵ2≤−∑i=1nμ7siγ2+1+μ7nηγ2ϵ2γ2+1.
Since0<γ2+1<2, the following inequality holds by using Lemma 2:(32)s=∑i=1nsi21/2≤∑i=1nsiγ2+11/γ2+1.
Substituting (32) into (31) yields(33)−∑i=1nμ7sisiγ2tanhsiϵ2≤−μ7sγ2+1+μ7nηγ2ϵ2γ2+1.
Substituting (30) and (33) into (29) results in(34)V˙≤−μ6s2−μ7sγ2+1−μ4−12−μ42kmin−μ42lminD˜2−μ3μ5s˜2−∑i=1nδ12−μ42lminθ˜f¯i2+r0,where r0=D∗2/2+μ7nηγ2ϵ2γ2+1+δ12/2∑i=1nθf¯i∗2. By choosing parameters μ4,lmin and δ1 satisfy(35)λD=μ4−12−μ42kmin−μ42lmin>0,λθF=δ12−μ42lmin>0,and let rmin=minμ6/2,λD/2,μ3/2,ηf¯1λθF,…,ηf¯nλθF. It can be obtained as(36)V˙≤−rminV+r0.
From (36), we know(37)V≤r0rmin+V0e−rmint.
Obviously, it can conclude that all signals in (23) are ultimately uniformly bounded. The proof is completed.Theorem 2.
Based on Theorem1 and the dynamic system of the integral sliding mode variable itself, the ultimate bound of e is estimated as e≤ϵ0/μ11−ς1/γ1+1.Proof.
Theorem1 shows that the designed sliding mode controller (21) guarantees that all signals in (23) are bounded. Now, we further study the convergence of the tracking error e. According to (14), (15), and (21), we have(38)e.=s.−μ1Tanhϵ1γ1e−μ2Tanhϵ10e=ψs,D˜,θ˜f¯1,…,θ˜f¯n−μ1Tanhϵ1γ1e−μ2Tanhϵ10e,where ψs,D˜,θ˜f¯1,…,θ˜f¯n=L−1θ˜FTφFx+D˜−μ6s−μ7Tanhϵ2γ2s. Through the conclusion of Theorem 1, there exists a positive constant ϖ such that(39)ψs,D˜,θ˜f¯1,…,θ˜f¯n≤ϖ.
Consider the following Lyapunov function:(40)V¯=12eTe.
The derivative ofV¯ is calculated as(41)V¯.=eTψs,D˜,θ˜f¯1,…,θ˜f¯n−μ1eTTanhϵ1γ1e−μ2eTTanhϵ10e≤ϖe−∑i=1nμ1eieiγ1tanheiϵ1−∑i=1nμ2eitanheiϵ1.
By using Lemma1 and 2, the following inequalities hold:(42)−∑i=1nμ1eieiγ1tanheiϵ1≤−∑i=1nμ1eiγ1+1+μ1nηγ1ϵ1γ1+1≤−μ1eγ+1+μ1nηγ1ϵ1γ1+1,(43)−∑i=1nμ2eitanheiϵ1≤−∑i=1nμ2ei+μ2nηγ0ϵ1≤−μ2e+μ2nηγ0ε1.
Substituting (42) and (43) into (41), we have(44)V¯.≤−μ2−ϖe−μ1eγ1+1+ϵ0,where ϵ0=μ1nηγ1ϵ1γ1+1+μ2nηγ0ϵ1. Selecting μ2 satisfies ϵ0=μ1nηγ1ϵ1γ1+1+μ2nηγ0ϵ1, and define the compact set as(45)Ωe=e∈Rn|μ1eγ1+1≤ϵ01−ς,where ς∈0,1 is any number. If e∉Ωe, then(46)V¯˙≤−μ1eγ1+1+ϵ0<−μ1ςeγ1+1=−2γ1+1/2μ1ςV¯γ1+1/2.
Obviously,V¯ decreases monotonically outside the set Ωe until it enters the minimal level set of V¯ containing Ωe.Remark 4.
Because parameterϵ1 can be selected as a small positive constant, which means that parameter ϵ0 is also very small, Ωe is a very small neighborhood.


## 3.1. Construction of Integral Sliding Mode and Disturbance Observer

In order to design a sliding mode robust controller to make the tracking errore stable, the integral sliding variable needs to be designed. In [25], an integral terminal sliding variable is designed as(13)s=e+∫0tμ1Signγ1edτ,where s=s1,s2,…,snT∈Rn, Signγ1e=e1γ1signe1,e2γ1signe2,…,enγ1signenT, γ1∈0,1, and μ1 is a positive parameter. By using the traditional sliding mode control method, the occurrence of chatter phenomenon is mainly due to the hysteresis of control switching. To avoid the chatter phenomenon, we modify the sliding surface (13) as follows:(14)s=e+∫0tμ1Tanhϵ1γ1e+μ2Tanhϵ10edτ,where Tanhϵ1γ1e=e1γ1tanhe1/ϵ1,e2γ1tanhe2/ϵ1,…,enγ1tanhen/ϵ1T, Tanhϵ10e=tanhe1/ϵ1,tanhe2/ϵ1,…,tanhen/ϵ1T, μ2 is the designed positive parameter and ϵ1 is a small positive constant, and γ1 and μ1 are the same as in (13).Remark 3.
In general, the aim of using the integral sliding variables in (13) is to improve its control performance and eliminate the reaching phase under the nominal control law. Different from the former, based on the dynamic system of integral sliding variable s in (14), this paper is to explore the ultimate boundary of the tracking error e.According to (10) and (14), the dynamic system of s is obtained as follows:(15)s.=L−1θ˜FTφFx+L−1θ^FTφFx+u+D−x.d+μ1Tanhϵ1γ1e+μ2Tanhϵ10e.In order to show the performance of RBFNNs, definings^=s^1,s^2,…,s^nT as the estimation of s, we have(16)s^.=−μ3s˜+L−1θ^FTφFx+u+D^−x.d+μ1Tanhϵ1γ1e+μ2Tanhϵ10e,where s˜=s^−s, μ3 is the positive design constant, and D^=D^1,D^2,…,D^nT is the estimation of D. Furthermore, the derivative of s˜ is obtained as(17)s˜.=−μ3s˜−L−1θ˜FTφFx−D˜.Now, the disturbance observerD^ is constructed as(18)D^=μ4s−w,w.=L−1θ^FTφFx+u+D^−x.d+μ1Tanhϵ1γ1e+μ2Tanhϵ10e+μ4−1−s+μ5s˜,where w=w1,w2,…,wnT∈Rn is an auxiliary variable vector and μ4 and μ5 are positive design parameters.The derivative ofD^ can be written as(19)D^.=μ4s.−w.=μ4L−1θ˜FTφFx+D˜+s−μ5s˜,where D˜=D−D^=D˜1,D˜2,…,D˜nT, and then, we obtain(20)D˜.=D.−s+μ5s˜−μ4L−1θ˜FTφFx+D˜.


## 3.2. Main Results

Now, the controllerut is designed as(21)u=−μ6s−μ7Tanhϵ2γ2s−L−1θ^FTφFx−D^+x.d−μ1Tanhϵ1γ1e−μ2Tanhϵ10e,and the parameter adaptive laws are proposed as(22)θ^.f¯i=ηf¯ili−1siφf¯ix−μ5li−1s˜iφf¯ix−δ1θ^f¯i,where Tanhϵ2γ2s=s1γ2tanhs1/ϵ2,s2γ2tanhs2/ϵ2,…,snγ2tanhsn/ϵ2T, δ1,μ6,μ7, and ηf¯i are the design parameters, i=1,2,…,n, and ϵ2 is a small positive constant. We give the main result as follows.Theorem 1.
For the error system (10) under Assumptions 1–4 and the disturbance observer (18), the controller (21) and parameter adaptive laws (22) guarantee that all the closed-loop system signals in (23) are uniformly ultimately bounded.Proof.
Consider the Lyapunov function as(23)V=V1+V2+V3+V4,where(24)V1=12sTs=12s2=∑i=1n12si2,V2=12D˜TD˜=12D˜2=∑i=1n12D˜i2,V3=μ52s˜Ts˜=μ52s˜2=∑i=1nμ52s˜i2,V4=∑i=1n12ηf¯iθ˜f¯i2=∑i=1n12ηf¯iθ˜f¯iTθ˜f¯i.
Substituting (21) into (15), the derivative of V1 can be obtained as(25)V˙1=sTs.=−μ6sTs−μ7sTTanhϵ2γ2s+sTD˜+sTL−1θ˜FTφFx=−μ6sTs−∑i=1nμ7sisiγ2tanhsiϵ2+sTD˜+∑i=1nsili−1θ˜f¯iTφf¯x.
From (20), the derivative of V2 yields(26)V˙2=D˜TD˜.=D˜TD.−μ4D˜TD˜−D˜Ts+μ5D˜Ts˜−∑i=1nμ4D˜ili−1θ˜f¯iTφf¯ix.
From (17), the derivative of V3 yields(27)V˙3=μ5s˜Ts˜.=−μ3μ5s˜Ts˜−μ5s˜TD˜−∑i=1nμ5li−1s˜iθ˜f¯iTφf¯ix.
From (22), the derivative of V4 yields(28)V˙4=∑i=1n1ηf¯iθ˜f¯iTθ˜.f¯i=∑i=1n−li−1siθ˜f¯iTφf¯ix+μ5li−1s˜iθ˜f¯iTφf¯ix+δ1θ˜f¯iTθ^f¯i.
Using (25), (26), (27) and (28), we have(29)V˙=−μ6sTs−∑i=1nμ7sisiγ2tanhsiϵ2+D˜TD.−μ4D˜TD˜−∑i=1nμ4D˜ili−1θ˜f¯iTφf¯ix−μ3μ5s˜Ts˜+∑i=1nδ1θ˜f¯iTθ^f¯i.
Since the following inequalities hold,(30)D˜TD.≤D˜22+D.22≤D˜22+D∗22,−μ4li−1D˜iθ˜f¯iTφf¯ix≤μ4p∗2D˜i22lmin+μ4θ˜f¯i22lmin,δ1θ˜f¯iTθ^f¯i≤−δ12θ˜f¯i2+δ12θf¯i∗2,where lmin=minl1,l2,…,ln.
By using Lemma1, one has(31)−∑i=1nμ7sisiγ2tanhsiϵ2≤−∑i=1nμ7siγ2+1+μ7nηγ2ϵ2γ2+1.
Since0<γ2+1<2, the following inequality holds by using Lemma 2:(32)s=∑i=1nsi21/2≤∑i=1nsiγ2+11/γ2+1.
Substituting (32) into (31) yields(33)−∑i=1nμ7sisiγ2tanhsiϵ2≤−μ7sγ2+1+μ7nηγ2ϵ2γ2+1.
Substituting (30) and (33) into (29) results in(34)V˙≤−μ6s2−μ7sγ2+1−μ4−12−μ42kmin−μ42lminD˜2−μ3μ5s˜2−∑i=1nδ12−μ42lminθ˜f¯i2+r0,where r0=D∗2/2+μ7nηγ2ϵ2γ2+1+δ12/2∑i=1nθf¯i∗2. By choosing parameters μ4,lmin and δ1 satisfy(35)λD=μ4−12−μ42kmin−μ42lmin>0,λθF=δ12−μ42lmin>0,and let rmin=minμ6/2,λD/2,μ3/2,ηf¯1λθF,…,ηf¯nλθF. It can be obtained as(36)V˙≤−rminV+r0.
From (36), we know(37)V≤r0rmin+V0e−rmint.
Obviously, it can conclude that all signals in (23) are ultimately uniformly bounded. The proof is completed.Theorem 2.
Based on Theorem1 and the dynamic system of the integral sliding mode variable itself, the ultimate bound of e is estimated as e≤ϵ0/μ11−ς1/γ1+1.Proof.
Theorem1 shows that the designed sliding mode controller (21) guarantees that all signals in (23) are bounded. Now, we further study the convergence of the tracking error e. According to (14), (15), and (21), we have(38)e.=s.−μ1Tanhϵ1γ1e−μ2Tanhϵ10e=ψs,D˜,θ˜f¯1,…,θ˜f¯n−μ1Tanhϵ1γ1e−μ2Tanhϵ10e,where ψs,D˜,θ˜f¯1,…,θ˜f¯n=L−1θ˜FTφFx+D˜−μ6s−μ7Tanhϵ2γ2s. Through the conclusion of Theorem 1, there exists a positive constant ϖ such that(39)ψs,D˜,θ˜f¯1,…,θ˜f¯n≤ϖ.
Consider the following Lyapunov function:(40)V¯=12eTe.
The derivative ofV¯ is calculated as(41)V¯.=eTψs,D˜,θ˜f¯1,…,θ˜f¯n−μ1eTTanhϵ1γ1e−μ2eTTanhϵ10e≤ϖe−∑i=1nμ1eieiγ1tanheiϵ1−∑i=1nμ2eitanheiϵ1.
By using Lemma1 and 2, the following inequalities hold:(42)−∑i=1nμ1eieiγ1tanheiϵ1≤−∑i=1nμ1eiγ1+1+μ1nηγ1ϵ1γ1+1≤−μ1eγ+1+μ1nηγ1ϵ1γ1+1,(43)−∑i=1nμ2eitanheiϵ1≤−∑i=1nμ2ei+μ2nηγ0ϵ1≤−μ2e+μ2nηγ0ε1.
Substituting (42) and (43) into (41), we have(44)V¯.≤−μ2−ϖe−μ1eγ1+1+ϵ0,where ϵ0=μ1nηγ1ϵ1γ1+1+μ2nηγ0ϵ1. Selecting μ2 satisfies ϵ0=μ1nηγ1ϵ1γ1+1+μ2nηγ0ϵ1, and define the compact set as(45)Ωe=e∈Rn|μ1eγ1+1≤ϵ01−ς,where ς∈0,1 is any number. If e∉Ωe, then(46)V¯˙≤−μ1eγ1+1+ϵ0<−μ1ςeγ1+1=−2γ1+1/2μ1ςV¯γ1+1/2.
Obviously,V¯ decreases monotonically outside the set Ωe until it enters the minimal level set of V¯ containing Ωe.Remark 4.
Because parameterϵ1 can be selected as a small positive constant, which means that parameter ϵ0 is also very small, Ωe is a very small neighborhood.


## 4. Example

In this section, the chaotic finance system [26] is introduced to demonstrate the effectiveness of the proposed control method (21) and the chaotic finance system is described as follows:(47)x˙1=x3−x1x2−0.8︸f1x+d1t+satu1,x˙2=1−x12−0.2x2︸f2x+d2t+satu2,x˙3=−1.9x3−x1︸f3x+d3t+satu3,where d1t=2.5sint, d2t=2.0cos2t, and d3t=2.5sin3t. The reference signal xd=sint,sint,sintT. Here, initial values of x,s˜, and w are chose as x0=2,−1,−2.5T, s^0=1.2,1.2,1.2T, and w0=−2,2,3T, and the parameters are chose as ui∗=10, li=5, ηfi=50, μ1=μ2=5,μ3=μ5=μ7=3,μ4=15,μ6=4, ϵ1=ϵ2=0.0005, ς=1/7,δ1=0.01, and i=1,2,3. The fuzzy membership functions are selected as(48)φfζ=exp−12ζ+7.5−2.5k1.22,where k=1,2,3,4,5;ζ=x1,x2,x3.According to Theorem2, the ultimate bound of the tracking error e can be estimated in the interval −2.4×10−3,2.4×10−3 when γ1=1/5. Now, we give the traditional integral sliding mode control method (TISMC) as follows:(49)s=e+∫0tμ1Signγ1edτ,u=−μ6s−μ7signγ2s−ϑ^F¯TφF¯x+x.d−μ1Signγ1e,ϑ^.F¯i=50siφF¯x−0.01ϑ^F¯i,where ϑ^F¯TφF¯x=ϑ^F¯1TφF¯x,ϑ^F¯2TφF¯x,ϑ^F¯3TφF¯xT is the estimation of Fx+Δu+dt. Obviously, the approximation range of e cannot be obtained directly from TISMC (49). In the following, TISMC (49) is discussed in three cases.(i)
Case 1:γ1=1/5,γ2=1/5.Figures1 and 2 show the simulation results for case 1. It can be seen that the tracking error quickly approaches zero, but the fluctuation range of the error e exceeds −2.4×10−3,2.4×10−3, and both the error state e=e1,e2,e3T and the saturation control input satu=satu1,satu2satu3T appear as chattering phenomenon.Case 2:γ1=1/8,γ2=1/5.Figures3 and 4 show the simulation results for case 2. It can be seen from Figure 3 that the convergence speed of the error e is the same as that of case 1, and the fluctuation range of the error e is limited to the interval −2.4×10−3,2.4×10−3. However, Figures 3 and 4 also reflect the chattering phenomenon of the error state e and saturation control input satu is more severe than that of case 1.Case 3:γ1=1/3,γ2=1/5.Figures5 and 6 show the simulation results for case 3. Figure 6 shows that the chattering phenomenon of the saturation control input satu is reduced, but it is found from Figure 5 that the effect of the error approaching zero is severely weakened compared to cases 1 and 2.The simulation results of cases 1–3 show that the parametersγ1 and γ2 in control method (49) have a direct influence on the control effect of the error e. In other words, the smaller the fluctuation range of the error e, the more violent the chattering phenomenon of the control input.Figure 1
Case 1: the trajectories of tracking errore1,e2,e3 by method (48) in time period: (a) [0, 20s]; (b) [10s, 20s].
(a)(b)Figure 2
Case 1: the trajectories ofsatu1,satu2, and satu3 by using method (48).Figure 3
Case 2: the trajectories of tracking errore1,e2,e3 by method (49) in time period: (a) [0, 20s]; (b) [10s, 20s].
(a)(b)Figure 4
Case 2: the trajectories ofsatu1,satu2, and satu3 by using method (49).Figure 5
Case 3: the trajectories of tracking errore1,e2,e3 by method (49) in time period: (a) [0, 20s]; (b)[10s, 20s].
(a)(b)Figure 6
Case 3: the trajectories ofsatu1,satu2, and satu3 by using method (49).By using the proposed control method (21) in this paper, simulation results are shown in Figures 7 and 8 when γ1=γ2=1/5. It can be seen from Figure 7 that e is completely within the interval −2.4×10−3,2.4×10−3, and it is found from Figure 8 that there is no chattering phenomenon in the saturation control input satu. Let e¯fi+Δui+di=li−1θ^f¯iTφf¯x+D^i−fix−Δui−dit and efi+Δui+di=ϑ^F¯iTφF¯x−fix−Δui−dit be the errors of system uncertainty by using methods (21) and (49), respectively. It can be seen from Figure 9 that the proposed control method (21) can better estimate system uncertainty than control method (49). Therefore, through simulation results, it is better to show that the proposed control method (21) has better control effect than method (49).Figure 7
The trajectories of tracking errore1,e2,e3 by method (21) in time period: (a) [0, 20 s]; (b) [10 s, 20 s].
(a)(b)Figure 8
The trajectories ofsatu1,satu2, and satu3 by using method (21).Figure 9
The trajectories of (a)e¯f1+Δu1+d1, ef1+Δu1+d1; (b) e¯f2+Δu2+d2, ef2+Δu2+d2; and (c) e¯f3+Δu3+d3, ef3+Δu3+d3 by using methods (21) and (49).
(a)(b)(c)


## 5. Conclusion

By using an integrated sliding mode control strategy and disturbance observer, this paper considers the tracking stability problem of a class of uncertain chaotic systems with saturation inputs. Firstly, the unknown functionFx is estimated by using fuzzy logic systems, and the external disturbance dt, the fuzzy estimation error L−1εFx, and the control error Δu=satu−u are regarded as an unknown compound disturbance D. Then, the integral sliding variable s is proposed, and the disturbance observer D^ is constructed by s and D. The proposed control strategy u and parameter adaptive rules θ^f¯i can ensure that all signals of the closed-loop system are bounded, and the ultimate bound of the tracking error can be estimated. Simulation results show that the proposed method in this paper can effectively control the tracking error quickly and eliminate the chattering phenomenon. Meanwhile, the system uncertainty Fx+Δu+dt can be effectively estimated.



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