Xuemei Zheng,Peng Li,Haoyu Li,and Danmei Ding

Department of Electrical Engineering,Harbin Institute of Technology,Harbin 150001,China

Adaptive backstepping-based NTSM control for unmatched uncertain nonlinear systems

Xuemei Zheng*,Peng Li,Haoyu Li,and Danmei Ding

Department of Electrical Engineering,Harbin Institute of Technology,Harbin 150001,China

An adaptive backstepping-based non-singular terminal sliding mode(NTSM)control method is proposed for a class of uncertain nonlinear systems in the parameteric-strict feedback form.The adaptive control law is combined with therst n?1 steps of the backstepping method to estimate the unknown parameters of the system.In the nth step,an NTSM control strategy is utilized to drive the last state of the system to converge in anite time.Furthermore,the derivate estimator is used to obtain the derivates of the states of the error system;the higher-order non-singular terminal sliding mode control(HONTSMC)law is designed to eliminate the chattering and make the system robust to both matched and unmatched uncertainties.Compared to the adaptive backstepping-based linear sliding mode control method (LSMC),the proposed method improves the convergence rate and the steady-state tracking accuracy of the system,and makes the control signal smoother.Finally,the compared simulation results are presented to validate the method.

adaptive control,backstepping,terminal sliding mode, higher-order sliding mode control,robustness.

1.Introduction

Slidingmodecontrol(SMC)is a nonlinearcontrolmethod. It alters the dynamics of a nonlinear system by using a discontinuous control signal and forces the system to slide along a predesigned sliding mode manifold.Compared to other control methods,SMC has attracted signicant interest among application engineers and researchers due to its simplicity in good performances,such as robustness to external disturbancesand low sensitivity to system parameter variations[1–8].Conventional SMC systems adopt linear sliding mode(LSM)controllers.The design task of the LSM controller includes the design of the sliding mode and control strategy.The former is based on the requirementsofthesystem performance.Thelatteris basedonthe existing condition of the sliding mode.

Based on LSM control,the terminal sliding mode (TSM)controller is developed[9–13].Compared to LSM manifold,TSM offers some superiorproperties,mainly including fast andnite-time convergence,and high steady state precision.However,the TSM controller has a singularity problem,that is,in some areas of the whole state space,the control for guaranteeing the TSM motion of the system may cause innity.The singularity problem limits the application of TSM control.A carefully designed switching scheme to avoid the singularity in TSM control systems was proposed in[14–16].The sliding-mode of the system switches between TSM and LSM.When a singularity appears,the sliding-mode switches from TSM to LSM;it switches back from LSM to TSM as soon as the system trajectory passes the singularity area.The disadvantage of this method is that the convergence time is extended.However,the chattering in LSM or TSM is always the problem that has to be overcome.

To avoid chattering,some approaches were proposed [17–20].The main idea is to change the dynamics in a small vicinity of the discontinuity surface in order to avoid real discontinuity and at the same time to preserve the main properties of the whole system.However,the ultimate accuracy and robustness of the sliding mode are partially lost.The recently invented higherorder sliding mode (HOSM)generalizes the basic sliding mode idea,acting onthehigherordertime derivativesofthesystem deviation fromtheconstraintinsteadofinuencingtherst deviation derivativeas it happensinstandardslidingmodes.Keeping the main advantages of the original approach,they totally remove the chattering effect and provide even higher accuracy in realization.HOSM is actually a movement on a discontinuity set of a dynamic system understood in Filippov’s sense.The sliding order characterizes the dynamics smoothnessdegreein the vicinityof themode.If thetask is to keep a constraint given by equality of a smooth function to zero,the sliding order is a number of continuous totalderivativesof the sliding mode in the vicinity of the sliding mode.Hence,the rth order sliding mode is determined by the equalities:

forming an r-dimensional condition on the state of the dynamic system.

The backsteppingstrategy is characterizedby a step-bystep procedure that interlaces,at each step,a coordinate transformation with the design of a virtual control via a classical Lyapunovtechnique,throughthe denition of the turning function.As a result,at the last step,the true control expression and the actual update law are obtained.

In this paper,an adaptive backsteppingbased TSM control method is proposed for a class of uncertain nonlinear systems in the parameteric-strict feedback form.The adaptive control law is combined with therst n–1 steps of the backstepping method to estimate the unknownparameters of the system.In the nth step,a higher-order nonsingular terminal sliding mode control(HONTSM)is then utilized to drive the last state of the system to converge in anite period of time.Furthermore,the derivate estimator is used to obtain the derivates of the states of the error system;the HONTSM control law is designed to eliminate the chattering and make the system robust to both matched and unmatched uncertainties.Compared to the adaptive backstepping-based LSM control,the proposed methodimprovesthe convergencerate and the steady-state trackingaccuracyof the system,andmakes the controlsignal smoother.Finally,the simulation results are presented to validate the method.The advantages of the proposed combined procedure are numerous:it allows the presence of nonparametric uncertainties in the last two equations of thesystem andincreases therobustness.Furthermore,it reducesthe computionalload,as comparedwith the standard backstepping strategy.

2.System description

Considerthefollowingstrictparameterfeedbacknth order uncertain nonlinear system:

where x = [x1,...,xn]T∈ Rnis the observability matrix,u and y ∈ R1is the system input and output respectively,f(x)is the nonlinear function,and b is the constant(b/=0).Δf(t)is the matched uncertainty,θTΦi(x1,x2,...,xi)(i=1,...,n?1)is the unmatched uncertainty,but can be expressed by a parameter. θ=[θ1,...,θm]T∈Rmis the uncertain constant matrix. Φi(x1,x2,...,xi)∈Rmis the known nonlinear smooth function.

Proposition 1The matched uncertainties and their derivates satisfy the following conditions:

where lfand lffare constants.

Proposition 2The uncertain parameter θ satises

where lθis a constant.

The paper control objective is:for system(1),if Proposition 1 and Proposition 2 are satised,design the adaptive backstepping HONTSM control law to adjust the output andlet y=ydwhereydis thedesiredsystem output,while for the matched uncertainty Δf(t)and the unmatched uncertaintyθTΦi(x1,x2,...,xi),the controllaw has the robustness under parameter uncertainty and disturbance.

3.Design of backstepping HONTSM controller

The paper is based on[3],designed by the following steps:

(i)System(1)is transformed into the error system;

(ii)Design the virtual controller for each subsystem and adjust the uncertain parameter so that the Lypunov function of each subsystem is negative;

(iii)Ensurethatthewholesystemis stableandhasbetter robustness.

Firstly,the paper adopts the adaptive control method to deal with the unmatcheduncertaintyfor the(n?1)th step; while in the nth step,the paper designs an HONTSM control law to deal with the matched uncertainty.Then the whole system possesses the robustness.

3.1Standard adaptive backstepping algorithm

The standard adaptive procedure is designed as follows from therst step to the(n?1)th step[21]:

where i=1,...,n?1,Γ=ΓT＞0 is the gained matrix,τiis the adaptiveadjustingfunctionωiis the designed parameter,and ci＞0 is a positive constant.

For system(7),the ith subsystem selects the following Lyapunov function:

After the above(n?1)th step designing,the n?1 virtual control ai(x1,...,xi,?θ)(i=1,...,n?1),can be gotten.Based on(8),this paper designs the HONTSM control method,and the following adaptive parameter adjust law and control law.

3.2HONTSM control method

According to(8),the Lyapunov function of the(n?1)th step is designed as follows:

Derivating(9)and considering(7),then we can get:

From(10),it can be seen that if,then

If we design the control law and make znconverge to zero,then

which can guarantee the subsystem of z2to zn?1is stable.

In order to let znconverge to zero innite time and speed up the convergence,this paper adopts the following HONTSM surface:

where p and q are odds and satisfy 1＜p/q＜ 2,and γ＞0.As for the relationship of p and q,the singularity cannot appear,so(11)is called NTSM.s is used to realize the second-order SMC,which can realize s=˙s=¨s=0, and can be called higher-order sliding mode and eliminate the chattering as for the higher order of s.If the system converges on tr,s converges to zero,which means s(t)=0,t≥tr,then from(11),it can be known that znand˙znwill converge to zero innite time,the time is

where t≥ts.The system will keep on the second-order sliding mode(zn= ˙zn=0),according to(12),and by selecting the parameters p and q,γ can adjust the convergence speed of zn.

This paper proposes the following theorem and designs the second-order NTSM control law.This method improves the standard adaptive backstepping algorithm.

Theorem 1If the system satises

ProofDerivate(10)as

Considering

then

Derivate the above equation,then

Considering

then

If s/=0,as p and q are odds and 1＜p/q＜2,then ˙zp/q?1n≥0,which means˙Vs≤0.If s/=0,we will consider the two following conditions:

(i)If zn/=0,then˙Vs≤?(rp/q)ip/q?1nη|s|＜0;

(ii)If˙zn=0,zn/=0,˙Vs=0 cannot be kept continuously[3].

Thus,according to the Lyapunov stable theorem,the system can be kept in the NTSM manifold innite time. After that,zncan be converged innite time.

After znconverges to zero,foris negative about the error of system(z1,...,zn?1,?θ)T.Thus the equivalent of the system (z1,...,zn?1,?θ)T=(0,...,0,θ)is stable on the whole. That is,z1,...,zn?1converges to the virgin of zero and?θ also converges to the virgin of zero.

4.Simulation

In order to testify the proposed control strategy,this paper adopts the linear SMC and HONTSM control respectively to the following system:

From(2)we can get Φ1(x1)=(x1+0.3)2,f(x)= x1sin(x2)+x2,g=1.Suppose uncertain parameter θ=0.5,and f(t)=20sin(4t).The upper of the uncertainty is lf=20,lff=80,lθ=1.The control law is to adjust y=x1and yd=2.Dene the error variable as z1=x1?yd,z2=x2?a1.The initial state of the system is x1(0)=x2(0)=0.

From(7)we can get

According to Theorem 1,design thefollowing HONTSM surface and control law:

where

Case 1Select parameter:c1=20,Γ=5,γ=0.02, p=5,q=3,η=120.

As Φe1=Φ1(yd)=(2+0.3)2/=0,rank(Φe1)=m =1,the original system is stable and?θ converges to θ=0.5.

Simulation results are shown in Figs.1–6.

Fig.1 System states x1and x2of Case 1

Fig.2 System states z1and z2of Case 1

Fig.3 Sliding mode s of Case 1

Fig.4 Phase plane of z2and˙z2and their integral of Case 1

Fig.5 Estimated valueof Case 1

Fig.6 Controlled variable u of Case 1

Case 2When yd=?0.3 is set,rank(Φe1)=0,and the other parameter is stable,the result of the simulation is shown in Figs.7–10.It can be seen that s=z1still converges asymptotically,but?θ cannot converge to θ=0.5.

Fig.8 Estimated value?θ of Case 2

Fig.9 Sliding mode s of Case 2

Fig.10 Phase plane of z2and˙z2and their integral of Case 2

IftheLSMis adoptedforthesamementionedcontrolled target,design the sliding mode as

The control law is

Case 3The choosen parameter:c1=5,Γ =5, δ1=2,k=2,λ=20,ε=10.

The result of simulation is shown in Figs.11–15.It can be seen that the speed is lower than the above simulation, and the chattering phenomenon exists in the sliding mode.

Fig.11 System states x1and x2of Case 3

Fig.12 System states z1and z2of Case 3

Fig.14 Controlled variable uof Case 3

5.Conclusions

For the class of uncertain nonlinear systems in the parameteric-strict feedback form,this paper proposes an adaptive backstepping-based HONTSM control method. The adaptive control law is combined with therst n?1 steps of the backstepping method to estimate the unknown parameters of the system.In the nth step,an HONTSM is then utilized to drive the last state of the system to converge in anite period of time.Compared to the adaptive backstepping-based LSM control method,the proposed methodimprovesthe convergencerate and the steady-state trackingaccuracyof the system,andmakes the controlsignal smoother.Finally,the simulation results are presented to validate the method.

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Fig.13 Sliding mode s of Case 3

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Biographies

Xuemei Zheng was born in 1969.She recieved her B.S.degree in electrical engineering from Northeast Heavy Machinery College,M.S.degree in control and application,and Ph.D.degree in electrical engineering from Harbin Institute of Technology.She is an associate professor and supervisor of postgraduate in Department of Electrical Engineering,Harbin Institute of Technology.Her research interests include nonlinear system control and wind power system control.

E-mail:xmzheng@hit.edu.cn

Peng Li was born in 1989.He received his B.S.degree in power electronics from Yanshan University in 2013.He is a master student in Department of Electrical Engineering,Harbin Institute of Technology.His research interests include nonlinear system control and wind power system control.

E-mail:pengliysuhit@163.com

Haoyu Li was born in 1974.He received his B.S. degree in power system,M.S.degree in electrical engineering,and Ph.D.degree in control theory and application from Harbin Institute of Technology in 1995,1997 and 2001,respectively.He is a professor and a supervisor of doctor.His research interests are power conversion and control technology in harsh environment.

E-mail:lihy@hit.edu.cn

Danmei Ding was born in 1991.She received her B.S.degree in electrical engineering from Harbin Institute of Technology in 2013.She is now a master student in Department of Electrical Engineering,Harbin Institute of Technology.Her research interests include nonlinear system control and wind power system control.

E-mail:ddm2285@163.com

10.1109/JSEE.2015.00063

Manuscript received April 28,2014.

*Corresponding author.

This work was supported by the Natural Science Foundation of Heilongjiang Province(E201426).