Zhou ZHENG,Weijie SUN,Hui CHEN,John T.W.YEOW

College of Automation Science and Engineering,South China University of Technology,Guangzhou Guangdong 510640,China

Integral sliding mode based optimal composite nonlinear feedback control for a class of systems

Zhou ZHENG,Weijie SUN,Hui CHEN,John T.W.YEOW?

College of Automation Science and Engineering,South China University of Technology,Guangzhou Guangdong 510640,China

This paper presents an optimization method of designing the integral sliding mode(ISM)based composite nonlinear feedback(CNF)controller for a class of low order linear systems with input saturation.The optimal CNF control is first designed as a nominal control to yield high tracking speed and low overshoot.The selection of all the tuning parameters for the CNF control law is turned into a minimization problem and solved automatically by particle swarm optimization(PSO)algorithm.Subsequently,the discontinuous control law is introduced to reject matched disturbances.Then,the optimal ISM-CNF control law is achieved as the sum of the optimal CNF control law and the discontinuous control law.The effectiveness of the optimal ISM-CNF controller is verified by comparing with a step by step designed one.High tracking performance is achieved by applying the optimal ISM-CNF controller to the tracking control of the micromirror.

Composite nonlinear feedback;Integral sliding mode;Input saturation;PSO;Micromirror

1 Introduction

Settling time and overshoot are two important transient performance indices.Most of nonlinear techniques for improving transient performance often make a tradeoff between them.Lin et al.[1]proposed the composite nonlinear feedback(CNF)technique to improve the tracking performance for a class of second order linear system with input saturation.The CNF control law is composed of a linear feedback law and a nonlinear feedback law,and it takes advantage of varying damping ratio.Chen et al.[2]developed the CNF control to a more general class of systems with measurement feedback and it was extended to multivariable systems in[3].A MATLAB toolkit is presented in[4]for the CNF control system design.In absence of disturbances,the CNF control technique has been successfully applied to the hard disk driver(HDD)system[2,5],the DC motorsystem[6],the gantry crane system[7]and the active suspension system[8].However,disturbances are inevitable in practical plant environment.

Sliding mode control is a robust control technique for linear and nonlinear systems[9].In the conventional sliding mode control,the system is not robust during the reaching phase.Integral sliding mode(ISM)concept is proposed in[10]to solve the reaching phase problem.The main idea of the ISM control is to define the control law as the sum of a continuous nominal control and a discontinuous control[11].The nominal control law achieves desired tracking properties for the nominal plant,while the discontinuous control law takes care of disturbances.LQR(linear quadratic regulator)control is taken as a nominal control in[12].Bandyopadhyay et al.[13]proposed integral sliding mode based composite nonlinear feedback(ISM-CNF)control,which combined advantages of the CNF control like quick response with negligible overshoot and robustness of the ISM control.The ISM-CNF control has been successfully applied to the servo position control system[14].

The performance of the ISM-CNF controller relies on the design procedure,especially on the design of the CNF control.However,it is always a big problem to select appropriate parameters for a CNF control law.Lan et al.[15]proposed a new nonlinear function and an auto-tuning method for the CNF control law and they have been successfully applied to the HDD system[5]and the DC motor system[6].However,the selection of the linear feedback gain is not addressed.Simple guidelines on selecting linear feedback gain are mentioned in[2,3].In[7],the particle swarm optimization(PSO)algorithm[16]is employed to solve the optimal control problem for the gantry crane system.However,the selection of the positive definite matrixWof Lyapunov equation which would affect the performance of the nonlinear part of the CNF control is not discussed.This motivates us to develop a completely auto-tuning method to simplify the design of the CNF control law,then combine it with the discontinuous control law to gain the ISM-CNF control law,which could track a reference target robustly with desired transient performance.In this paper,we fully take into account all the parameters to be adjusted of both the linear part and nonlinear part of the CNF control law,including the matrixW.The parameter tuning problem is converted into a performance criteria minimization problem.The integral of time multiplied absolute-value of error(ITAE)[17]is chosen as the criteria to evaluate the transient performance of the closed-loop system.The optimal CNF control law is designed by solving the minimization problem with the PSO algorithm.Then,the integral sliding surface and the discontinuous control law is designed following the design procedure presented in[11].The optimal ISMCNF control law is computed by adding up the optimal CNF control law and the discontinuous control law.Simulation results show that the optimal ISM-CNF control law achieves robustness,quick response and negligible overshoot.

2 Composite nonlinear feedback control

The CNF control law for a single input single output linear system will be reviewed in this section to show the concept in a simple way.The control objective is to drive the controlled output to track the target reference with high transient performance,in the presence of matched uncertainty and disturbance.

Consider a linear system with input saturation and matched uncertainty

wherex∈Rn,u∈R,y∈R are respectively the state,the control input and the controlled output of the system.A,BandCare constant matrixs with appropriate dimensions.d(x,t)is bounded matched uncertainty or disturbance and only bounds are known,and the maximum bound isdmax.Function sat:R→R represents the actuator saturation defined as

whereumaxis the saturation level of the input.Assumptions are made as follows:

A1:(A,B)is stabilizable;

A2:(A,B,C)is invertible and has no zeros ats=0.

The linear feedback law is first designed as

whereris the tracking target,Fis the linear feedback gain and required to make(A+BF)asymptotically stable,andGis given as

The desired statexeis calculated by

Let?xbe the error between actual state and desired state

Then,the nonlinear feedback law is designed as

where ρ(r,y)is a nonpositive nonlinear function and locally Lipschitz iny.Pis the positive definite solution to the Lyapunov equation

for a given positive definite matrixW.

The CNF control law is derived as the sum of the linear feedback law and the nonlinear feedback law

Remark 1The nonlinear function ρ(r,y)is used to change the damping ratio of the closed-loop system as the controlled output approaches the tracking target.One choice of ρ(r,y)is given in[2]as

wherey0=y(0)and β>0 is a tuning parameter.

Function(10)works well with a given reference target,but it can't adapt the variation of tracking targets.In order to solve this problem,Lan et al.[15]developed the nonlinear function to the form of

where α >0 and β>0 are tuning parameters,and

The new nonlinear function is invariant to the tracking targets and has been successfully used in[5]and[6].Its effectiveness will be shown in Section 5.

Remark 2For any δ∈ (0,1),letcδ>0 be the largest positive scalar meeting the condition that for allthere exists

In presence of matched disturbances,both the linear control law(3)and the CNF control law(9)can drive the controlled output to asymptotically track the command input under the condition that the initial statex0andrsatisfy[13]

whereand|(δ-

3 Integral sliding mode control

For system(1),the integral sliding mode control law[11]is given in the form of

The nominal controlu0is responsible for tracking properties of the nominal plant and it can be of any form such as PID control,LQR control[12]and so on.In this paper,the CNF control law(9)is designed as the nominal control.The discontinuous controlu1takes care of the matched uncertainty and disturbance by ensuring the sliding motion.The integral sliding surface proposed in[11]is

whereG1is selected asso as to avoid amplification of unmatched perturbation.In(16),x(t)is the actual trajectory,andis the desired trajectory caused by nominal control in absence of uncertainty and disturbance.Therefore,s(x,t)can be considered as the difference between the actual and the desired trajectory projected onG1.It is observed that the system always starts at the sliding manifold,which means the reaching phase is eliminated.The objective is to find an appropriate discontinuous controlu1to keep the sliding motion ons(x,t)=0.The usual choice for the discontinuous control is[11]

whereM(x,t)≥dmaxis a gain related to the maximum bound of disturbances.In order to reduce chattering,the discontinuous control is approximated by

where θ is a very small positive number and set as 0.001 in this paper.Thus,the ISM-CNF control law can be derived from equation(9),(18)and(15)

Fig.1 shows the block diagram of the ISM-CNF control law.By defining Lyapunov function asV=?xTP?x,stability of closed-loop system(1)with control input(15)is proved for three cases in[13].

Fig.1 The block diagram of the ISM-CNF based control law.

4 Optimal control problem

In order to obtain desired tracking performance and robustness of the closed-loop system,parameters and feedback gains in the ISM-CNF control law need to be selected properly.The design of the discontinuous control law is relatively simple,because the only parameter needs to be tuned isM(x,t),which is determined by the maximum bound of disturbances.Therefore,the main problem is to design the linear feedback gainF,the positive definite matrixWof the Lyapunov equation and the nonlinear function ρ(r,y).

4.1 Minimization problem

There are several ways to determineF,such as H2and H∞approaches,pole placement method,LQR method and so on.Chen et al.[2]presents a guideline for the selection ofWby defining a auxiliary system.But the guideline is not operational in the design procedure.Generally,the matrixWas well as the parameters of the nonlinear function ρ(r,y)are tuned by trial and error method.The tracking performance of the closed-loop system mainly relies on the designer's experience.In order to obtain the desired tracking performance,Yu et al.[7]investigated the optimal CNF control problem.In[7],both the linear feedback gainFand the nonlinear function ρ(r,y)are determined automatically by solving the minimization problem with PSO algorithm.However,the selection of the positive definite matrixWof the Lyapunov equation is still not discussed,and disturbance is not considered either.This motivates us to develop a new optimal problem,which takesWand disturbance into account.Select properF,α,β andWsuch that

with the restrictions that

R1:A+BFis Hurwitz,

R2:α >0,β>0,

R3:initial damping ratio 0<ζ0<1,and

R4:for any δ ∈ (0,1),letcδ>0 be the largest positive scalar meeting the condition that

where 0 ≤ δ1< δ and|(δ-δ1)umax|=dmax.

Note that other restrictions can be extended easily according to practical situations.For example,|y|<|r|should be a restriction for systems that do not allow overshoot,and the state trajectory can be restricted in a given space.Moreover,we only take ITAE as the performance criteria in this paper,it could be replaced by other criteria such as IAE as well[17].PSO algorithm is employed to solve the minimization problem.After the optimal CNF control law is tuned,combine it with the discontinuous control law to form the optimal ISM-CNF control law so as to achieve the desired properties.

4.2 PSO algorithm

Particle swarm optimization(PSO)is a population based stochastic optimization technique which simulates the behaviors of bird flocking.In the basic PSO algorithm,the particle swarm consists ofnparticles,and the position of each particle stands for the potential solution in theD-dimensional space.In every iteration,each particle is updated by following two 'best' values.The first one is the best solution it has achieved so far,called personal best fitness.Another one is global best fitness,the best value obtained so far by any particle in the population.

Each particle can be shown by its current velocity and position,the personal best fitness and the global best fitness.The position and velocity of each particle are calculated as follows:

wherei=1,2,...,n,nis the particle number of the particle swarm andDis the dimension number of the searching space,tis the iteration time step,ω is the inertia weight,c1andc2are acceleration constants,xiD(t)andviD(t)are respectively the position and velocity of theith particle in iterationt,piDrepresents the best position in dimensionDachieved so far by theith particle,pgDis the best position in dimensionDachieved by the neighbours of theith particle,andr1andr2are random values uniformly distributed in[0,1].

PSO is easy to implement and there are few parameters to adjust.It has been successfully applied in many areas:function optimization,artificial neural network training,fuzzy system control,and other areas.In this paper,PSO is used to solve the minimization problem(20).

Consider the second-order linear system,then the linear feedback gainFand the positive definite matrixWare respectively set as

Therefore,there are six parameters to be tuned:f1,f2,w1,w2,α and β.We would like to point out that the order of systems in our research is no more than four.

5 Simulation results

Two examples are demonstrated in this section.Example 1 verifies the validity of the optimal ISM-CNF controller by comparing with the ISM-CNF controllerin[14].Example 2 shows application of the optimal controller in the tracking control of the micromirror.

5.1 Example 1

Consider the servo position control system with disturbance presented in[14]

wherex1,x2,uare respectively the angular position of the load disc,the angular velocity of the load disc and the input voltage.The maximum available input voltage isumax=4.9V,dis the disturbance and taken as

To design the optimal ISM-CNF controller for system(25),an optimal CNF controller should be designed first for the nominal system(d=0)by solving the minimization problem(20)with PSO algorithm.The reference step signal to be tracked is 50.The linear feedback gainFand the positive definite matrixWare respectively set as(23)and(24).By makingw1=w2,Wcan be expressed as

where η∈[-8,8]is a parameter to be tuned.Therefore,the tuning parameters are reduced tof1,f2,η,α and β.The conditions of the particle swarm in(21)are initialized as follows:n=1000,ω=0.9,c1=c2=1.2,the iteration is 30.

The tuning parameters are achieved as

Thus,

where the positive definite matrixP?is obtained as the solution to the Lyapunov equation(8).The optimal CNF controller can be achieved by putting the tuned parameters into equation(9).

Then,the discontinuous control lawu1is designed as follows:G1is computed byG1=B+=[0 0.0022]and the sliding surface is obtained by(16).Then,Mis chosen as 0.21 according to the maximum value of the disturbance.

The optimal ISM-CNF control is expressed as equation(19)with the designed parameters.

Fig.2 shows comparison of the controllers in[14]and the optimal controllers when the reference signal is set asr=50.It can be observed that the optimal controllers achieve much higher tracking performance in the presence of disturbance.

Fig.2 Comparison of the controllers in[14]and the optimal controllers when r=50.

The nonlinear function(10)is applied in the CNF control law and the ISM-CNF control law in[14].In this paper,we employ the nonlinear function(11)to adapt to the variation of the tracking target.Its effectiveness is tested by changing the tracking target intor=10 and still using the optimal ISM-CNF control law(19).

It is observed in Table 1 that the optimal controllers yield quicker response with smaller overshoot in all cases.Fig.3 shows that the CNF control law in[14]causes large oscillation and fails to ensure the stability of the perturbed system,while the optimal CNF control law takes advantage of the nonlinear function(11)and achieves high tracking performance.It can also be seen that the optimal ISM-CNF control law achieves robustness,quick response and negligible overshoot,while the ISM-CNF control law in[14]only attenuates the disturbance and still gives large overshoot.

Table 1 Comparison of the control law in[14]and the optimal control law.

Fig.3 Comparison of the controllers in[14]and the optimal controllers when r=10.

5.2 Example 2

The optimal ISM-CNF control can be applied to the tracking control of the 2D torsional micromirror with sidewall electrodes,which has been widely adopted in imaging systems,optical switches and adaptive optics.The dynamic equations of the electrostatically actuated micromirror system are expressed as[18]

where the parameters are given asR1=0.16,R2=0.15,λαβ=0.2251,Gα=3.0827X106,Gβ=1.7894X107,and the initial values isx(0)=(0;0;0;0).The control objective is to drive the scan angles α and β to track the target references with desired dynamic performance.TαandTβare electrostatic torques determined by the driving voltages.The simulation is simplified with approximate treatment by defining the control inputsuα=106TαandThe maximum available inputs are.Thus,the dynamic equations of the system can be seen as two independent systems:system(28)is for the control of α and system(29)is for the control of β.In the simulation,the disturbance is defined asd=0.3sin(10t).

Consider system(28),the target reference of the scan angle α isrα=1?.The parameters of the CNF control law are tuned by PSO algorithm and achieved asThe discontinuous control law is designed withMα=0.31 andGα1=[0 0.3244].Hence,the optimal ISM-CNF control lawis derived as equation(19)with

The optimal ISM-CNF controllers are tested in simulations.Fig.4 shows the tracking performance of the closed-loop system for the scan angle α and β respectively.It is observed that both the optimal ISM-CNF controllers suppress the effect of the disturbance and give desired tracking performance.More precisely,under the designed controllers,the overshoot is 0.49% and settling time is 0.32ms for the tracking of α,for β,there is no overshoot while the settling time is 0.69ms.The plot of control inputs with the optimal ISM-CNF control law are shown in Fig.5.The inputs are sinusoidally varying due to sinusoidal disturbance.

Fig.4 Tracking performance with the optimal ISM-CNF controllers.

Fig.5 Inputs with the optimal ISM-CNF controllers.

6 Conclusions

In this paper,a systematic method is proposed to design the integral sliding mode based optimal composite nonlinear feedback control law.First,the design of the optimal CNF control law is turned into a minimization problem with some restrictions and solved by PSO algorithm.Subsequently,the discontinuous control law is substituted by a continuous control law so as to reduce chattering.Then,the optimal ISM-CNF control law is achieved as the sum of the optimal CNF control law and the approximate continuous control law.Simulation results show that the optimal ISM-CNF control law yields much better tracking performance in comparison with the ISM-CNF control law in[14].The optimal controller is applied to the tracking control of the micromirror and achieves high tracking speed and negligible overshoot in presence of disturbance.

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21 February 2014;revised 28 March 2014;accepted 31 March 2014

DOI10.1007/s11768-014-0022-4

?Corresponding author.

E-mail:jyeow@uwaterloo.ca.Tel.:+86-13826040831;fax:+1 519-746-4791.

This work was supported by National Natural Science Foundation of China(No.61374036)and the Fundamental Research Funds for the Central Universities(No.SCUT 2014ZM0035).

?2014 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Zhou ZHENGreceived his B.S.degree in Automation from Wuhan University of Technology in 2012.He is currently working toward his M.S.degree at South China University of Technology.His main research interests include modeling and control of micro-electro-mechanical system.E-mail:weasonlife@126.com.

Weijie SUNis a lecturer in South China University of Technology.His research interests include control theory and applications,robotics and automation,dynamical analysis and design of micro/nano devices.E-mail:auwjsun@scut.edu.cn.

Hui CHENreceived his B.S.degree in Automation and M.S.degree in Control Theory and Control Engineering,both from the Henan Polytechnic University in 2002 and 2007,respectively.He received a scholarship from China Scholarship Council and was a visiting scholar in the Advanced Micro/Nano Devices Lab.at University of Waterloo from 2011 to 2013.He is a Ph.D.candidate at South China University of Technology.His current research interests include control and applications of micro/nano devices.E-mail:huichenscut@gmail.com.

John T.W.YEOWreceived the B.S.degree in Electrical and Computer Engineering,and M.A.S.and Ph.D.degrees in Mechanical and Industrial Engineering from the University of Toronto,Toronto,ON,Canada,in 1997,1999,and 2003,respectively.He is currently a professor in the Department of Systems Design Engineering at University of Waterloo,Waterloo,ON,Canada.His current research interests are in the field of developing miniaturized biomedical instruments.He is a Canada Research Chair in Micro/Nanodevices.He is the Editor-in-Chief of the IEEE Nanotechnology Magazine,and an Associate Editor of the IEEE Transactions on Nanotechnology.E-mail:jyeow@uwaterloo.ca.