Backstepping sliding mode control for uncertain strictfeedback nonlinear systems using neural-network-based adaptive gain scheduling

YANG Yuenengand YAN Ye

College of Aerospace Science and Engineering,National University of Defense Technology,Changsha 410073,China

1.Introduction

It is well known that the backstepping control technique has become one of the most effective control approaches for nonlinear systems in strict-feedback form[1–4].The key feature of backstepping design is that it stabilizes the system states through a step-by-step recursive process,which reduces the complexity of the control design and guarantees the Lyapunov stability of the closed-loop system[5–7].

For more general and challenging problems,involving system uncertainties and external disturbances which are difficult to deal with,the above-mentioned backstepping control approach may not achieve the control goal.With respect to this problem,the sliding mode control(SMC)represents an attractive alternative,which has a particular property of insensitivity to parametric uncertainty and external disturbance due to the use of a switching control law[8–13].Therefore,the back-stepping sliding mode control(BSMC)approach which combines both the merits of backstepping control and SMC,is proposed to address the control problem of uncertain strict-feedback nonlinear systems.However,discontinuous control in SMC is not realizable via the backstepping approach based on Lyapunov stability theory employing the so-called “hard”gain control laws,which yields undesirable chattering[14,15].To decrease the chattering,a radial basis function neural network(RBFNN)is employed to construct the A neural network-based adaptive gain scheduling backstepping sliding mode control(NNAGS-BSMC)to facilitate adaptive gain scheduling,in which the gains are scheduled adaptively via neural network(NN),with sliding surface and its differential as NN inputs and the gains as NN outputs[9,16].Thus,the proposed control approach,notonly takes advantage of design flexibility of backstepping control,but also brings into full play SMC’s robustness against parametric uncertainties and external disturbances.

The rest of this paper is organized as follows.In the next section,the control problem is formulated.Section 3 proposes BSMC and NNAGS-BSMC for uncertain strict feedback nonlinear systems.The verification example is presented to demonstrate the effectiveness of proposed control approach in Section 4.Finally,conclusions are given in Section 5.

2.System description and problem formulation

Consider the following uncertain strict-feedback nonlinear system[17,18]:

whereand y∈R are the state variables,system input and output,respectively,which are all assumed to be available for measurement;f(x)and g(x)are smooth nonlinear functions and d(x,t)includes parametric uncertainties and external disturbances.

Assumption 1For the referenced signal yr,it is assumed that yris bounded and with n−1 bounded derivatives,i.e.,there exists a positive constant Y＞0 such that

where‖·‖denotes the Euclidean norm.

Assumption 2The parametric uncertainties and external disturbances are assumed to be bounded such that

where D is a positive constant.

Remark 1In order to design the BSMC,the Assumption 2 given in(3)is employed.

The control problem of the current paper is stated as:given any initial states x1(0)and x2(0),and the referenced signal yr,consider the nonlinear system defined by(1),design an appropriate law such that the system output can track the referenced signal asymptotically,i.e.,

where yrdenotes the referenced signal,e is the tracking error.

3.Control design

3.1 BSMC

The current subsection presents a BSMC approach for the control problem[17–19].The detailed design is described step-by-step as follows.

Algorithm 1BSMC

Inputs:

i)referenced output yr;

ii)system output y.

Output:the control input u for system(1).

Step 1Design of virtual control α1

i)Compute the tracking error e1=y−yr;

ii)Select the optional parameter k1＞0;

iii)Design the virtual control α1=−k1e1+˙yr;

iv)Compute the derivative of e1:

Step 2Design of virtual control α2

i)Compute the tracking error e2=x2−α1;

ii)Select the optional parameter k2＞0;

iii)Design the virtual control

iv)Compute the derivative of

Step iDesign of virtual control αi

i)Compute the tracking error ei=xi− αi−1;

ii)Select the optional parameter ki＞0;

iii)Design the virtual control

iv)Compute the derivative of

Step nDesign of the control law

i)Compute the tracking error

ii)Select the optional parameter c＞0;

iii)Define the following sliding surface s=cen−1+en;

iv)Select the optional parameters ρ ＞ 0,λ ＞ 0;

v)Design the reaching law

vi)Design the control input u:

Step n+1Termination

If the tolerance of control error is satisfied,terminate the algorithm and output u.Otherwise,go to Step 1.

The stability of the closed-loop system is proven via the following theorem.

Theorem 1For the nonlinear system(1),if the sliding surface is chosen as

and the control input is designed as

then it guarantees that all the signals of the closed-loop system are bounded.

ProofSelect the following Lyapunov function candidate:

Differentiating(7)with respect to time and using the derivatives of eiand s,it is obtained that

Substituting(6)into(8)yields

Based on Assumption 2,we have

By selecting the optional gain,λ satisfies the following condition:

and we have

Define μ =min{2k1,...,2(kn−c),2ρ},and then(9)can be expressed as follows:

Remark 2It is proven that all states and signals are bounded,and it is concluded that the BSMC designed as(6)ensures Lyapunov stability of the nonlinear system(1).

3.2 NNAGS-BSMC

The designed BSMC provides an effective control approach for the system(1).However,after the system state reaches the sliding phase,chattering phenomena is brought about near the sliding surface,that is the major drawback of the BSMC.Because the control law(6)includes the sign function,the repeated switching of the sliding surface induces chattering,and the gain of sign function determines the chattering intensity.To address this problem,an RBFNN is adopted to construct the NNAGS-BSMC to facilitate adaptive gain scheduling in this subsection(its block diagram is depicted in Fig.1),in which the gains are scheduled adaptively,with sliding surface s and its differential˙s as NN inputs and the gain λ as NN output.

Fig.1 Block diagram of NNAGS-BSMC

The design of RBFNN is described in detail as follows[20–26].

Algorithm 2RBFNN

Inputs:

i)sliding surface s;

ii)its differential˙s.

Output:gain λNof sign function.

Step 1Defining the input layer

The sliding surface and its differential are defined as NN net inputs,i.e.,s=[s,˙s]T;

Step 2Selecting the membership function

The following membership function is selected:

where μiand σiare the mean and the standard deviations of the Gaussian function in the i th term,respectively.

Step 3Calculating the output

Calculate the gain of the sign function:

where wiis the weight connecting the i th RBF μiand the output,and n is the number of RBFs.

Step 4Design of learning algorithm

i)Select the energy function

where e=y−yris the tracking error;

ii)Design the learning algorithm based on the back propagation method:

Step 5Termination

If the chattering is decreased and the tolerance of control error is satisfied,terminate the algorithm and output λN.Otherwise,go to Step 1.

Remark 3The mean and standard deviations of the kernel layer are updated in the similar way.The gain λNcan be scheduled adaptively via RBFNN with switching sliding surface and its differential as inputs.

Based on the designed RBFNN,the NNAGS-BSMC is designed as follows:

4.Verification example

In the current subsection,simulation results are presented to illustrate the effectiveness and robustness of the designed control approaches.Take the following nonlinear system for instance:

The referenced output is selected as yr=0.5 sin(π/15t),the external disturbance is selected as 0.02 sin(π/100t),and the initial state is x1(0)=0.035 and x2(0)= 0.Simulations are performed by using BSMC,BSMC using saturation function(SF-BSMC)and NNAGS-BSMC,respectively.

4.1 Simulation of BSMC

The optional parameters of BSMC are chosen as ρ=50,λ=15.Simulation results of BSMC are shown in Fig.2.Fig.2(a)shows the referenced output(dashed)and system output(real line).The system output y with an initial value of 0.035 increases to the referenced output yrwithin 2 s.Fig.2(b)shows the control errors.The error decreases to zero after 2 s.Fig.2(a)and Fig.2(b)indicate that BSMC is effective for tracking the referenced output.Fig.2(c)displays the control input of BSMC.The control input switches continuously,and the chattering phenomenon is obvious due to the use of a switching control law.

Fig.2 Simulation results of BSMC

4.2 Simulation of SF-BSMC

Replacing the sign function with a saturation function is one of the most common solutions to the chattering problem.The sign function of BSMC given by(6)is replaced with a saturation function,and the SF-BSMC is expressed as follows:

The other optional parameters of SF-BSMC are the same as those of BSMC.Simulation results of SF-BSMC are shown in Fig.3.Fig.3(a)shows the referenced output(dashed)and system output(real line),and Fig.3(b)shows the control errors.Similar to Fig.2(a)and Fig.2(b),the system output tracks the referenced output effectively,and the error decreases to zero after 2 s.Fig.3(c)displays the control input of SF-BSMC.It is shown that the chattering of control input has been decreased,which illustrates the performance of chattering decrease of the SF-BSMC.

Fig.3 Simulation results of SF-BSMC

4.3 Simulation of NNAGS-BSMC

The following Gaussian function is adopted to construct the RBFNN:

where μ =1.5,σ =0.25.

The optional parameters are chosen as ρ =15,γ =0.8.Different from BSMC and SF-BSMC,the gain of the sign function is not a constant,and it is scheduled online.Simulation results of NNAGS-BSMC are shown in Fig.4.Fig.4(a)shows the referenced output(dashed)and system output(real line),and Fig.4(b)shows the control errors.Similar to Fig.2(a)and Fig.2(b),the system output tracks the referenced output effectively,and the error decreases to zero after 2 s.Fig.4(c)displays the control input of NNAGS-BSMC.Compared with BSMC,the chattering of control inputs has been decreased effectively.Fig.4(d)displays the gain of the sign function,which is scheduled via the RBFNN adaptively.

Fig.4 Simulation results of NNAGS-BSMC

Remark 4i)The BSMC,SF-BSMC and NNAGSBSMC all address the control problem of strict-feedback nonlinear system with parametric uncertainties and external disturbances effectively.ii)The chattering is the major drawback of BSMC.iii)The SF-BSMC reduces the chattering by replacing the sign function with the saturation function.iv)The gain of NNAGS-SMC is scheduled adaptively with the sliding surface and its differential according to the designed RBFNN,which reduces the chattering effectively.

5.Conclusions

This paper has successfully demonstrated the development and application of the NNAGS-BSMC for uncertain strict feedback nonlinear systems with parametric uncertainties and external disturbances.The BSMC is designed firstly,then,an RBFNN is used to design the NNAGS-BSMC for adaptive self-gain-scheduling to decrease the chattering results from the sign function.Through the comparing results,it demonstrates that the NNAGS-BSMC decreases the chattering effectively and has better control performances against BSMC.

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