# Adaptive Second-Order Sliding Mode Control Design for a Class of Nonlinear Systems with Unknown Input

Authors: You Zheng; Jianxing Liu; Xinyi Liu; Dandan Fang; L. Wu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

An adaptive second-order sliding mode controller is proposed for a class of nonlinear systems with unknown input. The proposed controller continuously drives the sliding variable and its time derivative to zero in the presence of disturbances withunknown boundaries. A Lyapunov approach is used to show finite time stability for the system in the presence of a class of uncertainty. An illustrative simulation example is presented to demonstrate the performance and robustness of the proposed controller.


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## Body

## 1. Introduction

Sliding mode control (SMC) has gained much attention due to its attractive characteristics of finite time convergence and robustness against uncertainties [1–3]. Sliding mode control has been thoroughly studied, from both practical and theoretical point of view, for example, successfully applied for control of time delay systems [4, 5], control and observation of fuel cell power systems [6–11], control and observation of power converters [12–16], and adaptive control of flexible spacecraft [17, 18]. Robustness of a control system is essential based on the reason that various uncertainties exist in practical systems. These nonlinear systems are with structured and unstructured uncertainties and external disturbances such as load variation. However, the price of using sliding mode control to achieve robustness to these disturbances is control chattering problem [19–22]; chattering is undesired because it may cause high frequency dynamics and even instability. Therefore, in order to achieve the optimal operation performance of such uncertain systems, suitable model and controller development efforts are needed [23–26].There are several ways to avoid chattering problem when using sliding mode control. The conventional SMC uses a control law with large control gains yielding the undesired chattering while the control system is in the sliding mode. To eliminate the chattering, the discontinuous control function is replaced by “saturation” or continuous “sigmoid” functions in [27, 28]. However, such approach constrains the sliding system's trajectories not to the sliding surface but to its vicinity losing the robustness to the disturbances. Higher order sliding mode control techniques are used in [29–32] which allow driving the sliding variable and its consecutive derivatives to zero in the presence of the disturbances. However, the main challenge of high order sliding mode controllers is the use of high order time derivatives of sliding variable. It is interesting to note that the popular super-twisting algorithm [1] only requires the measurement of the sliding mode variable without its time derivative. Recently, adaptive sliding mode controller has been proposed to tune the controller gains with respect to disturbances [33, 34]. Intelligent controllers such as fuzzy neural network [35, 36] and adaptive fuzzy sliding mode controller [37, 38] were proposed to reduce the chattering. However, all the aforementioned literatures address the case based on the assumption that the boundary of the disturbances is known. This boundary cannot be easily obtained in practical cases. The overestimating of the disturbance boundary yields to larger than necessary control gains, while designing the super-twisting control law [32].In this paper, an adaptive controller gain super-twisting algorithm (ASTW) is proposed for nonlinear system with unknown input. Based on the Lyapunov theory, the proposed control law continuously drives the sliding variable and its time derivative to zero in the presence of bounded unknown input but without knowing the boundary. The stability and the robustness of the control system are proven, and the tracking performance is ensured.


## 2. Problem Formulation

The super-twisting control law (STW) is effective to remove the chattering when the relative degree equals one. It generates the continuous control function that drives the sliding variable and its time derivative to zero in finite time in the presence of bounded unknown disturbance. The main disadvantage of STW algorithm is that it requires the bounded value ofσ˙ [39] (σ is the sliding variable). Unfortunately, the knowledge of σ˙ is often unavailable. The overestimating of the σ˙ will yield to larger than necessary control gains while designing the STW control law. In this note, an adaptive-gain approach will be adopted when designing STW control law, which does not require the bounded value of σ˙. The idea of adaptive-gain approach is to increase the control gain α, β dynamically until the STW controller converges. Once the σ>0, then gains will start reducing; this gains reduction will be reversed as soon as the sliding variable σ and its time derivative σ˙ start deviating from the equilibrium point σ=σ˙=0.Consider a class ofnth-order uncertain nonlinear system, which is represented in a state-space form as(1)x˙=fx+bxu+d,where x=[x1,x2,…,xn]T=[x1,x˙1,…,x1n-1]T∈Rn is the state vector, u∈R is the control function, d∈Rn is the external disturbance, and f(x)∈Rn and b(x)∈Rn are the smooth vector fields.Let the desired state vector bexd; the tracking error and the sliding-surface function are defined as(2)ex,t=x-xdσx,t=cTe,where c=[c1,c2,…,cn]T. The sliding mode will be obtained in finite time if an appropriate control law is applied. In the sliding mode, the error dynamics will be(3)cne1n-1+cn-1e1n-2+⋯+c1e1=0.The constants c1,c2,…,cn are chosen to be positive such that the eigenvalue polynomial φλ=cnλn-1+cn-1λn-2+⋯+c1 is Hurwitz. The choice of c decides the convergence rate of the tracking error.Assumption 1.
The relative degree of system (1) with the sliding variable σ with respect to u equals one; in other words, the control function u has to appear explicitly in the first total derivative σ˙.Under Assumption1, the dynamics of σ can be computed as follows: (4)σ˙=gxu+ρx,t,where ρ(x,t)=cTf(x)+cTd-cTx˙d and g(x)=cTb(x)≠0.Assumption 2.
The first-order time derivative of the uncertain functionρ(x,t)∈R is bounded (5)ρ˙x,t≤δ,for someunknown constants δ>0.Assumption 3.
The uncertain functiong(x,t)∈R is presented as(6)gx,t=g0x,t+Δgx,t,where g0(x,t)>0 is the nominal parts of g(x,t) and Δg(x,t) is the parameter variation such that(7)Δgx,tg0x,t≤η<1.With Assumptions2 and 3, (4) can be written as follows: (8)σ˙=ρx,t+g1x,tω,where 1-η≤g1(x,t)=1+Δg(x,t)/g0(x,t)≤1+η and ω=g0(x,t)u. The solution of (8) is understood in the sense of Filippov [40].


## 3. Adaptive-Gain STW Controller Design

The control objective is to drive the sliding variableσ and its derivative σ˙ to zero in finite time without the control gain overestimation based on Assumptions 2 and 3 satisfied. The classical SMC can handle with the task to keep an output variable σ at zero when the relative degree of σ is one. However, the high frequency control switching leads to the chattering effect which is exhibited by high frequency vibration of the controlled plant and can be dangerous in some applications. The second-order sliding mode (SOSM) controllers including the continuous STW control algorithm are able to remove the chattering effect while preserving the main sliding mode features and improving its accuracy in the presence of unknown disturbance. However, it requires the knowledge of bounded disturbance. Unfortunately, the assumption of the disturbance is often unavailable in practice which leads to controller gain overestimation.In this note, an adaptive-gain approach is used to solve this problem with STW algorithm. STW controller generates the continuous control function to remove the chattering effect while adaptive-gain approach allows controller gain nonoverestimation.The STW algorithm can be described by differential inclusion as follows [1, 32]: (9)ω=-λσ1/2sign⁡σ+vv˙=-αsignσ,where the time varying controller gains(10)λ=λσ,t,α=ασ,tare to be defined later.Substituting (9) into (8), it follows that (11)σ˙=-λg1x,tσ1/2signσ+φ,φ˙=-αg1x,tsignσ+ϕx,t,where φ=g1(x,t)v+ρ(x,t) and ϕ(x,t)=g˙1(x,t)v+ρ˙(x,t).The termg˙1(x,t)v is assumed to be bounded with some unknown constant δ1; that is, (12)ϕx,t≤δ1+δ=δ2,for someunknown constant δ2>0.In this paper, the gainsλ(σ,t) and α(σ,t) are formulated as (13)λσ,t=λ0lt,ασ,t=α0lt,with λ0, α0 arbitrary positive constants and a positive time varying scalar l(t). The dynamic law of the varying function l(t) is given by(14)l˙tk,ifσ≠00.otherwise,where k>0 is a positive constant.Now, the control objective is reduced to drivingσ and its derivative σ˙ by (13) and (14) to zero in finite time with the condition of bounded perturbations in (12). Thus, the design of adaptive STW controller is formulated in the following theorem.Theorem 4.
Consider system (11); suppose that (12) holds. Then, for any initial conditions x(0), σ(0), the adaptive-gain STW control laws (13), (14) drive the sliding variable σ and its derivative σ˙ to zero in finite time.Proof.
A new state vector is introduced in order to present system (11) in a more convenient form for Lyapunov analysis:(15)ζ=ζ1ζ2=l1/2tσ1/2signσφ;thus, system (11) can be rewritten as(16)ζ˙=l2ζ1-λ¯01-2α¯00︸Aζ+0ϕx,t+l˙t2ltζ10,where λ¯0=λ0g1(x,t) and α¯0=α0g1(x,t). Given that g1(x,t)>0, it is easy to verify that A is a Hurwitz matrix.
Then, the following Lyapunov function candidate is introduced for system (16):(17)V=ζTPζ,P=124α¯0+λ¯02-λ¯0-λ¯02.
Taking the derivative of (17),(18)V˙=lt2ζ1ζTATP+PAζ+q1ϕx,tζ+l˙t2ltζ1q2ζ,where q1=-λ¯0,2 and q2=4α¯0+λ¯02,-λ¯0.
SinceA is a Hurwitz matrix, there exists a positive definite matrix Q such that ATP+PA=-Q and λmin⁡(P)ζ2≤V≤λmax⁡(P)ζ2. Equation (18) can be rewritten as(19)V˙≤-lt2λmin⁡Qλmax⁡1/2PV1/2+σ2q1λmin⁡1/2PV1/2+ΔV,where(20)ΔV=l˙t2lt4α¯0+λ¯02ζ12-2λ¯0ζ1ζ2≤l˙t2lt4α¯0+λ¯02+λ¯02ζ12+λ¯02ζ22=l˙t2ltζTΔQζ,ΔQ=4α¯0+λ¯02+λ¯0200λ¯02.With (20), (19) is present as(21)V˙≤-lt2λmin⁡Qλmax⁡1/2P-σ2q1λmin⁡1/2PV1/2+l˙t2ltλmax⁡ΔQλmin⁡PV.For simplicity, we define(22)γ1=λmin⁡Q2λmax⁡1/2P,γ2=σ2q1λmin⁡1/2P,γ3=λmax⁡ΔQ2λmin⁡P.γ1, γ2, γ3 are all positive constants. Thus, (21) is simplified as(23)V˙≤-γ1lt-γ2V1/2+γ3l˙tltV,where l˙≥0.
In order to show that theV˙ will be negative in finite time, the second time derivative of V is calculated as(24)V¨≤-l˙tV1/2-γ1lt-γ2V˙2V1/2+γ3l˙tltV˙-γ3l˙tl2tV.If V˙>0, the positive term γ3l˙(t)/l(t)V˙ is decreasing and the term (γ1l(t)-γ2)(V˙/2V1/2) will be positive at some time instant t=t1 which dominates the positive term γ3(l˙t/lt)V˙, since l(t) is a monotonic nondecreasing function (l˙(t)≥0). In this point, we can conclude that after the time t=t1, V¨<0. That is to say V is not increasing faster than a time linear function during time t∈[0,t1] which can be formulated as(25)V≤mt+n.
Substituting (25) into (23), we have(26)V˙≤-γ1lt-γ2V1/2+γ3l˙tmt+nkt+l0,where l(0)>0 is the initial value of the scalar function (14).
The positive term in (26) is bounded,(27)γ3l˙tmt+nkt+l0≤γ3l˙tmax⁡⁡m,nmin⁡⁡k,l0.
Now, we can conclude that after some timet>t1 the first term in the right side of (26) will dominate the second term such that (28)V˙≤-θγ1lt-γ2V1/2,with θ∈(0,1) and γ1l(t)-γ2>0. Therefore, σ and σ˙ converge to zero in finite time. Theorem 4 is proven.Remark 5.
In view of practical implementation, the condition|σ|=0 in (14) cannot be satisfied due to measurement noise and numerical approximations. The condition |σ|=0 needs to be modified by dead-zone technique [41, 42], such that the dynamic law (14) is practically implementable,(29)l˙t=k,ifσ≥τ0,else,where τ is a sufficiently small positive value.


## 4. Simulation Results and Discussions

Consider the following nonlinear system [43]: (30)x˙1=x2,x˙2=x3,x˙3=-b1tx12-b2tx2-b3tx3+3+cos⁡⁡tu+dt,where b1(t)=1+0.3sin(t), b2(t)=1.5+0.2cos⁡(t), b3(t)=1+0.4sin(t), and d(t)=0.05sin(0.1t) is considered as an external disturbance.The desired state trajectory is supposed to bexdT=[sin(t),cos⁡(t),-sin(t)]. The sliding-surface function (2) is chosen, and cT=[10,5,1]. The initial state vector is xt0=[0,0,-0.35]T. The initial value of the sliding variable is taken as σ(0)=-5.35.According to (2), the sliding surface is given as (31)σ=c1x1-x1d+c2x2-x2d+c3x3-x3d.The time derivative ofσ is calculated as (32)σ˙=g1x,tu+ρx,t,where g1(x,t)=c3(3+cos⁡t) and ρ(x,t)=-cTxd+c1x2+c2x3+c3d(t)+c3b1(t)x12-b2(t)x2-b3(t)x3.The adaptive-gain STW control laws (9), (10), (11), and (13) are designed as (33)σ˙=-λσ,tg1x,tσ1/2sign⁡σ+φφ˙=-ασ,tg1x,tsignσ+ϕx,t,where the adaptive-gain λ(σ,t) and α(σ,t) dynamics follow (29) and the values of the parameters of the adaptive-gain law have been taken as τ=0.005, λ0=2.5, α0=8, and k=5. These parameters are tuned to get sufficiently accurate and fast convergence. Finally, the adaptive-gain dynamic law (29) becomes(34)l˙t=5,ifσ≥0.0050,else.Figures1–3 show the state performance of system (3) in the presence of disturbance d(t). The time response of the sliding surface function σ is shown in Figure 4. It can be easily found that σ converges to zero in finite time. From Figure 5, we can see that the control input is smooth which is suitable for real applications. Figure 6 shows that the adaptive law of (34) is effective in the presence of external disturbance d(t).Figure 1
The performance of statex1 and its error e1.Figure 2
The performance of statex2 and its error e2.Figure 3
The performance of statex3 and its error e3.Figure 4
Sliding surface function of (31).Figure 5
Control inputu versus time(s).Figure 6
Time varying functionl(t).Remark 6.
In order to avoid high frequency control activity, the boundary layer technique is employed during the simulation. Thus, the saturation functionsat⁡σ/Φ is used to replace the function signσ/Φ, where Φ is the boundary layer width; that is,(35)sat⁡σΦ=σΦ,ifσΦ≥1signσΦ,else.


## 5. Conclusions

Adaptive second-order sliding mode control design is studied for a class of nonlinear systems with unknown inputs. In real applications, the upper boundary of uncertainty is difficult to obtain, which is required for calculating control gains of super-twisting sliding mode control. To solve this problem, a simple control design method is proposed based on Lyapunov function. The idea is very simple; the gains are increased according to a dynamic law until the sliding mode is attained and stops increasing thereafter. The proposed approach has two advantages:(i)
Only one parameterk has to be tuned.(ii)
A priori knowledge of the uncertainty bound is not required.



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