# Adaptive Chattering-Free Sliding Mode Control of Chaotic Systems with Unknown Input Nonlinearity via Smooth Hyperbolic Tangent Function

Authors: Jiunn-Shiou Fang; Jason Sheng-Hong Tsai; Jun-Juh Yan; Shu-Mei Guo
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The design of adaptive chattering-free sliding mode controller (SMC) for chaotic systems with unknown input nonlinearities is studied in this paper. A smooth hyperbolic tangent function is utilized to replace the discontinuous sign function; therefore, the proposed adaptive SMC ensures that not only the chaos phenomenon can be suppressed effectively but also the chattering often appearing in the traditional discontinuous SMC with sign function is eliminated, even when the unknown input nonlinearity is present. A sufficient condition for stability of closed-loop system is acquired by Lyapunov theory. The numerical simulation results are illustrated to verify the proposed adaptive sliding mode control method.


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## Body

## 1. Introduction

In recent decades, sliding mode control is one of the popular control theories due to its robustness and insensitivity to parameter uncertainty and external disturbance [1, 2]. The trajectories of SMC systems can be driven onto a specified sliding surface and slide toward desired stable condition, which is called sliding motion. There are many methodologies used to select sliding surfaces, while the different control law is obtained, and it is guaranteed that the controlled system is stable [2]. In general, the SMC and switched system control strategies are usually adopted to the practical application. For example, in [3, 4], the new SMC-based fuzzy controller and robust hybrid controller are applied to the 7-degree-of-freedom upper-limb exoskeleton robot such that the undesired external disturbance can be suppressed. Therefore, the SMC can be used to overcome undesired external disturbance successfully; in [5], a class of linear switched system is discussed for the cooperative stabilization problem and a novel class of switching signals is proposed. Based on above descriptions, the SMC is adopted to cope with input nonlinearity considered in this paper due to its robustness.In traditional SMC, the control force is difficult to achieve, which has high frequency chattering phenomenon due to the discontinuous sign function. Therefore, many quasisliding mode control techniques have been proposed [2, 6–8], and the chattering phenomenon is eliminated due to the controller designed by using a continuous function. On the other hand, chaotic systems have been proven the existence which is extensively and become an interesting topic in physical systems [9–11]. Also, the chaotic systems have special nonlinear dynamic behavior [7, 12]. In general, there are two sides to such physical systems [11]; the first side is the beneficial feature, such as chemical reactions and encrypted communication systems. On the other side, the chaos phenomenon in some engineering systems is highly unexpected for its applications, such as electronic systems, power converters, and high-precision mechanical systems. Therefore, many different methods have been proposed to solve the problems of chaos control and suppression for chaotic systems, such as adaptive feedback control [13–16], sliding mode control [11, 17, 18], PID control [19–21], optimal control [22–26], and robust control [27–29] among many others [30, 31].As described above, many modified methodologies have been developed to overcome the chattering phenomenon in traditional SMC, but they have not been well discussed when the system structure is subjected to input nonlinearity and even the input nonlinearity is unknown. It is well known that in practical systems, unavoidable external nonlinear perturbations do exist in control input, which not only affect system stability but also decrease the input performance and response. Furthermore, since the feature of chaotic system is very sensitive to parameter uncertainty and external perturbation [11], the stability analysis of controlled system with input nonlinearity is an important issue for chaotic systems. Recently, some papers [11, 32, 33] have researched SMC design for the chaotic systems with nonlinear input, but they only discussed the control design when input nonlinearity is well known. Hu et al. [34] proposed a SMC-based adaptive controller to deal with nonlinear input, but the adaptive law is high order. However, we present a low-order adaptive law to omit the harmful effects of nonlinear input. To our knowledge, the chattering-free SMC design with unknown input nonlinearity is still not well discussed.In this paper, the adaptive continuous SMC design for chaotic systems with unknown input nonlinearity is studied. The main contribution is to introduce a new technique of adaptive sliding mode control with a smooth hyperbolic tangent function to avoid chattering. In addition, the undesired chaotic behavior of the considered chaotic systems can be fully suppressed even with unknown input nonlinearities. Finally, we present numerical simulation results to illustrate the effectiveness of the proposed adaptive continuous SMC scheme.


### 1.1. Notations

In this paper,Rn represents the n-dimensional Euclidean space, Rn×m denotes the set of all real n by m matrices, the superscript “+” denotes the matrix generalized inverse, and Im stands for m by m identity matrices. W denotes the Euclidean norm, when W is a vector, and the induced norm, when W is a matrix. λmaxW denotes the maximum eigenvalue of matrix W. W represents the absolute value of W, and signs is the sign function of s; if s > 0, signs=1; if s = 0, signs=0; and if s < 0, signs=−1.


## 1.1. Notations

In this paper,Rn represents the n-dimensional Euclidean space, Rn×m denotes the set of all real n by m matrices, the superscript “+” denotes the matrix generalized inverse, and Im stands for m by m identity matrices. W denotes the Euclidean norm, when W is a vector, and the induced norm, when W is a matrix. λmaxW denotes the maximum eigenvalue of matrix W. W represents the absolute value of W, and signs is the sign function of s; if s > 0, signs=1; if s = 0, signs=0; and if s < 0, signs=−1.


## 2. System Description and Problem Formulation

Consider a general class of chaotic systems given below:(1)x˙t=Ax+fx,t,where A∈Rn×n is the system matrix and x∈Rn is the system state vector. fx,t∈Rn×1 denotes the nonlinear vector of systems. Without loss of generality, we make the following assumption about system (1).Assumption 1.
The dynamic system (1) can be written as(2)x˙t=Ax+Bgx,t,where B∈Rn×m and gx,t∈Rm×1. The pair A,B is controllable.Remark 1.
Assumption 1 is not restrictive. Many nonlinear chaotic systems described by (2) can be found. For example, the Matsumoto–Chua–Kobayashi circuit, Ro¨ssler system, modified Chua’s circuit, Lorenz system, Duffing–Holmes system, Lu¨ system, and Chen chaotic dynamical system.
To suppress the chaos oscillation behavior, we introduce a control vector subjected to unknown nonlinearity described as(3)x˙t=Ax+Bgx,t+Bφut,where φut=φ1u1⋯φmumT∈Rm×1 is the control vector with nonlinearity. The continuous nonlinear function satisfies φ0=0, where ϕiuit∈R⟶R with the law uit⟶φiuit, satisfying(4)β2,iui2≥uiφiui≥β1,iui2,where β1iandβ2i are unknown positive nonzero constants but bounded. Figure 1 shows a nonlinear function φiuit inside the sectors β1iandβ2i.
In SMC, the traditional controller is difficult to implement because there is an important adverse problem of high-frequency chattering phenomenon. In order to improve this issue, the following lemma with smooth hyperbolic tangent function is introduced.Figure 1
The scalar nonlinear functionφiuit.Lemma 1.
There always exists a constantγi>0 for all si≠0 such that(5)si⋅tanhsi>γisi⋅signsi.Proof.
Sincetanhsi=ejsi−e−jsi/ejsi+e−jsi,j>0, obviously, si⋅tanhsi>0 and si⋅signsi=si>0 for all si≠0. In general, there always exists a positive constant εi∈0,1 satisfying(6)sisignsi<sitanhsi+εisisignsi.
Also, it can be rewritten as(7)sitanhsi>1−εisisignsi=γisisignsi,where γi=1−εi. Therefore, there always exists a constant γi>0 satisfying si⋅tanhsi>γisi⋅signsi for all si≠0.
It is worthy to mention that the existing but unknown parameterγi does not appear in our proposed controller due to the adaptive control approach. By Lemma 1, a smooth continuous switching function is used to construct our control design such that the chattering phenomenon can be improved. As shown in Figure 2, the different parameters j are discussed for hyperbolic tangent function.
According to Figure2, the hyperbolic tangent function with a large value of j is close to the discontinuous sign function while the chattering phenomenon might appear. Therefore, the parameter j should be assigned by an appropriate value for avoiding the undesired chattering in SMC.Figure 2
Thetanhsi=ejsi−e−jsi/ejsi+e−jsi function with different j.


## 3. The Control Design Algorithm

In consequence, to complete the control objective mentioned above, there are two major steps. First, one needs to design an appropriate switching surface for the control system with input nonlinearity such that the stability of the dynamics on the sliding manifold defined later can be ensured. Second, one needs to propose a continuous adaptive SMC such that the existence of the sliding motion can be guaranteed without chattering.First, the proportional-integral (PI) type sliding surface is first defined as follows:(8)St=Cxt−∫0tCA+Kxτdτ,where St∈Rm,C∈Rm×n,andK∈Rm×n and matrix C=B+=BTB−1BT results in CB=Im. B+ is the generalized inverse of B, and B is the full-column rank. Im∈Rm is the identity matrix. The control matrix K satisfies λmaxA+BK<0. As long as the system can operate in the sliding mode, the controlled dynamics will satisfy the following equation:(9)S˙t=0,St=0.Therefore, when the system operates in the sliding mode, we can obtain the equivalent controlφequt by differentiating (8) with respect to time and substituting from (3):(10)S˙t=−Kxt+gx,t+φequt=0,where CB=B+B=Im has been introduced. Obviously, the equivalent control φequt in the sliding mode is obtained by(11)φequt=Kxt−gx,t.Substituting (11) into (3), we have(12)x˙t=A+BKxt.According the above discussion and (12), we can conclude that when the system is in the sliding manifold, the controlled chaotic system in the sliding mode is stable if matrix K satisfies λmaxA+BK<0. Now, to guarantee the existence of the sliding mode, the continuous sliding controller and adaptive law are proposed, respectively, as(13)ut=−αη^tgx,t−Kx+δtanhS,(14)η^˙t=αgx,t−Kx+δ⋅S.From (13) and (14), we have(15)uit=−αη^tgx,t−Kx+δtanhsi,η^˙t=αgx,t−Kx+δ⋅S,where α>1,δ>0 are constants and can be assigned. The factor η^t is used to estimate the unknown input nonlinearity. The control block diagram is shown in Figure 3.Figure 3
The control block diagram.Theorem 1.
Consider dynamic system (3) with unknown input nonlinearity. If the continuous adaptive SMC is properly designed as (13) and (14), then the system trajectory will be controlled to the sliding surface St=0 even with unknown input nonlinearities.Proof.
Let us consider a Lyapunov function for a closed-loop system as follows:(16)Vt=12STS+ηe2,where η=γ1⋅β1, β1=mini∈1,mβ1,i, γ1=mini∈1,mγi, e=η^t−η−1, and e˙=η^˙t and η is an unknown constant. Then, the Lyapunov function (16) derivative with respect to time is obtained as(17)V˙t=STS˙+eηe˙=STgx,t−Kx+φu+eηe˙≤S⋅gx,t−Kx+δ+STφu+eηe˙.
Sinceβ2,iui2≥uiφiui≥β1,iui2, one has(18)−αη^tgx,t−Kx+δtanhsiφui≥β1,iα2η^t2gx,t−Kx+δ2tanh2si,and then(19)−tanhsiφui≥αβ1,iη^tgx,t−Kx+δtanh2si.Sincesi2≥0, γ1=mini∈1,mγi, β1=mini∈1,mβ1,i, and(20)si⋅tanhsi>γisi⋅signsi,for allsi≠0,by substituting (20) into (19), we get the following result:(21)−tanhsisi2φui≥αβ1,iη^tgx,t−Kx+δsi2tanh2si,−siφui≥αβ1,iη^tgx,t−Kx+δsitanhsi,−siφui≥αγiβ1,iη^tgx,t−Kx+δsi⋅signsi,−siφui≥αγ1β1η^tgx,t−Kx+δsi⋅signsi.Letη=β1γ1 and ∑i=1msi≥S; then, we have(22)−siφui≥αηη^tgx,t−Kx+δ⋅si,thus(23)−∑i=1msiφui≥αηη^tgx,t−Kx+δ⋅∑i=1msi≥αηη^tgx,t−Kx+δ⋅S.Furthermore,∑i=1msiφui=STφu, and one has(24)STφu≤−αηη^tgx,t−Kx+δ⋅S.By substituting (13), (14), and (24) into the derivative of Lyapunov function (16), one has(25)V˙t≤S⋅gx,t−Kx+δ−αηη^tgx,t−Kx+δ⋅S+eηe˙=1−αS⋅gx,t−Kx+δ−αη^t−η−1ηgx,t−Kx+δ⋅S+eηη^˙t=1−αgx,t−Kx+δ⋅S.Letwt=1−αgx,t−Kx+δ⋅S; then, one has(26)Vt≤V0−∫0twλdλ,⟹V0≥Vt+∫0twλdλ≥∫0twλdλ.Thus, by using Barbalat lemma [31], we obtain limt⟶∞wt=0. Furthermore, since α>1,gx,t−Kx+δ>0, S⟶0 as t⟶∞. Hence, the proof is achieved completely.Remark 2.
Since a continuous adaptive SMC with a smooth hyperbolic tangent function is obtained, there is no high-frequency switching operation in sliding mode controller and the chattering is removed.


## 4. Numerical Simulations

In this section, to verify the proposed controller, we give two illustrative examples.Example 1.
Consider modified Chua’s circuit, and the dynamic system can be described as [35](27)x˙1t=px2t+p7x1t+fx1,t,x˙2t=x1t−x2t+x3t,x˙3t=−qx2t,with the nonlinear function(28)fx1,t=−2pm0x13,where p>0,q>0,m0 are the system parameters and xt=x1x2x3T is the state vector. The chaos response of Chua’s system without control force is shown in Figure 4. It is easy to verify that system (27) with nonlinear control input ϕut can be represented as(29)x˙=Ax+Bgx,t+ϕut,where A=p/7p01−110−q0, B=100, gx=−2pm0x13, p=10, q=100/7, and m0=1/7.Figure 4
The chaos response of modified Chua’s circuit without control force.The initial state condition is given byx0=−1.20.2−2T. The controller parameters are given by j=1, δ=8, α=1.2, and η^0=0. Let K=−6.4286−1.7143−4.58 such that the eigenvalues of the system are placed on −2,−3,−1. Also, for simulation, the unknown nonlinear function is given as(30)φut=1+0.3sinutut.Under the above-mentioned input nonlinearity, the corresponding state responses, the proposed controller (13), the sliding surface (8), and adaptive parameter are shown in Figures 5–8, respectively. In order to compare the proposed approach with the traditional SMC, the tanh function is replaced by the traditional sign function in the controller (13). Based on traditional SMC, the corresponding state responses, the control input, the sliding surface, and adaptive parameter are shown in Figures 9–12, respectively. Comparing Figures 5–8 with Figures 9–12, it is shown that the undesired chattering phenomenon can be fully suppressed by the proposed adaptive chattering-free control law (13).Figure 5
The system state response with the proposed continuous controller.Figure 6
The control input response with the proposed continuous control input.Figure 7
The switching function response with the continuous control input.Figure 8
The time response for the adaptive parameter with the proposed continuous controller.Figure 9
The system state response with the traditional SMC controller.Figure 10
The control input response with the traditional SMC controller.Figure 11
The switching function response with the traditional SMC controller.Figure 12
The time response for the adaptive parameter with the traditional SMC controller.Example 2.
Consider the chaotic Lorenz system [36] described as(31)x˙1t=−m1x1t+m1x2t,x˙2t=m2x1t−x2t+f1x,t,x˙3t=−m3x2t+f2x,t,where mi,i=1,2,3 are the system parameters and xt=x1x2x3T are the system states. The chaos response of the Lorenz system without control force is shown in Figure 13. Also, the state space with control input ϕut can be rewritten as (32)x˙=Ax+Bgx,t+ϕut,where A=−m1m10m2−1000−m3, B=001001, gx,t=f1x,tf2x,tT=−x1tx3tx1tx2tT, m1=10, m2=28, and m3=8/3.Figure 13
The chaos response of Lorenz system without control force.The initial state condition is given byx0=0−1.50.67T. The controller parameters are given by j=8, δ=8, α=3, and η^0=0. Let K=−34.370000.6667, and the eigenvalues of the system are placed on −2,−3,−1. Also, the nonlinear function is defined as(33)φut=1+0.3sinu2tut.Under the input nonlinearity, the corresponding state responses, the proposed controller (13), the sliding surface (8), and adaptive parameter are shown in Figures 14–17, respectively. To compare the proposed approach with the traditional SMC, the tanh function is also replaced by the traditional sign function in the controller (13). Based on traditional SMC, the corresponding state responses, the control input, the sliding surface, and adaptive parameter are shown in Figures 18–21. Comparing Figures 14–17 with Figures 18–21 shows that the undesired chattering phenomenon is suppressed by using the proposed control law (13); hence, the results demonstrate the validity of proposed method.From the simulation results above, it is concluded that the proposed method is effective and the chattering can be eliminated due to the adaptive continuous SMC even when the controlled systems are subjected to unknown input nonlinearities.Figure 14
The system state response with the proposed continuous controller.Figure 15
The control input response with the proposed continuous control input.Figure 16
The switching function response with the proposed switching function.Figure 17
The time response of Example 2 for the adaptive parameter with the proposed continuous controller.Figure 18
The system state response with the traditional SMC controller.Figure 19
The control input response with the traditional SMC controller.Figure 20
The switching function response with the traditional SMC controller.Figure 21
The time response for the adaptive parameter with the traditional SMC controller.


## 5. Conclusions

This paper has proposed a continuous adaptive SMC design for chaos suppression of a general class of chaotic systems. In contrast to the previous works, the type of continuous adaptive SMC with smooth hyperbolic tangent function is newly introduced such that not only the chaos of systems can be suppressed but also the chattering in conventional SMC can be eliminated even with unknown input nonlinearity. Numerical simulations have verified the effectiveness of the proposed method.



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