Zhihui WANG,Junyong ZHAI,Shumin FEI

Key Laboratory of Measurement and Control of CSE,Ministry of Education,School of Automation,Southeast University,Nanjing Jiangsu 210096,China

Global sampled-data output feedback control for a class of feedforward nonlinear systems

Zhihui WANG,Junyong ZHAI?,Shumin FEI

Key Laboratory of Measurement and Control of CSE,Ministry of Education,School of Automation,Southeast University,Nanjing Jiangsu 210096,China

This paper investigates the problem of global output feedback stabilization for a class of feedforward nonlinear systems via linear sampled-data control.To solve the problem,we first construct a linear sampled-data observer and controller.Then,a scaling gain is introduced into the proposed observer and controller.Finally,we use the sampled-data output feedback domination approach to find the explicit formula for choosing the scaling gain and the sampling period which renders the closed-loop system globally asymptotically stable.A simulation example is given to demonstrate the effectiveness of the proposed design procedure.

Sampled-data control;Global asymptotic stabilization;Output feedback;Feedforward nonlinear systems

1 Introduction

In this paper,we consider the problem of sampleddata output feedback stabilization for a class of feedforward nonlinear systems described by

whereandu(t)∈R are the system state,output and control input,respectively.φiis an unknown continuous function with φi(t,0,...,0)=0,i=1,...,n-1.The control law is implemented in discrete-time under a sampler and zero-order hold device.Our objective is to design a sampled-data controlleru(tk)only using discrete-time measurementy(tk),that is,

with the time instantstkandtk+1being the sampling points andTbeing the sampling period,to globally asymptotically stabilize system(1).

Feedforward systems(1),which are also called uppertriangular systems,are an important class of nonlinear systems.Since the pioneering work of Teel[1],many important stabilization results have been proposed for feedforward nonlinear systems,see,e.g.,[2-5]and the references therein.The work[2]studied the disturbance attenuation problem for a class of nonlinear feedforward systems.The work[3]solved the problem of stabilization for a class of nonlinear feedforward systems with bounded signals.The work[4]proposed a global highgain scaling state feedback controller for a general class of nonlinear systems containing uncertain functions of all the states and the control input.A small gain theorem was presented that provides a formalism for analysis the behavior of control systems in[5].In[6],a linear output feedback controller was constructed under a linear growth condition.Later in[7],a high-order growth condition was generalized by employing the homogeneous domination approach.The work[8,9]discussed the problem of global stabilization for feedforward nonlinear systems with an unknown output function.

More and more controllers are implemented using digital computers in practice,see,e.g.,[10-14].Compared with the elegant results obtained in the linear case[15],the problem of using sampled-data controllers to stabilize nonlinear systems is not a trivial one.There are several approaches to design of digital controllers.One main approach is to design a sampled-data controller based on the emulation method by discretizing continuous-time controllers[14,16,17].The issue in the emulation method is the choice of the sampling periodT.In the past serval years,the problem of output feedback sampled-data control of nonlinear systems has been studied in[11,12,16].With different type of discrete-time high-gain observers being designed,semi-global or local output feedback stabilization could be achieved for some nonlinear systems by choosing a small enough sampling period and sufficiently large enough observer gain.In[18],based on the technique of sampled-data feedback domination,a sampled-data state feedback controller was designed to globally asymptotically stabilize for a class of feedforward nonlinear systems.Then,a sampled-data output feedback controller is constructed to globally stabilize the nonlinear systems whose nonlinear terms are in lower-triangular form in[19].In this paper,we aim to deal with the problem of global sampled-data output feedback control for a class of feedforward nonlinear systems.By using a linear observer,a linear dynamic sampled-data output feedback control law is explicitly constructed.Then,a scaling gain is introduced into the observer and controller.Finally,it is shown that the closed-loop system can be globally stabilized by the proposed sampled-data output feedback controller with appropriately choosing the scaling gain and the sampling period.

2 Main results

In this section,we show that the problem of global output feedback stabilization for system(1)can be solved.The unknown functions φi(.),i=1,...,n-1 satisfy the following assumption.

Assumption 1There exists a positive constantcsuch that fori=1,...,n-1

Remark 1As shown in[6],the problem of global stabilization of system(1)is solvable under Assumption 1 by continuous-time controller.In this notes,we shall prove that system(1)can be globally stabilized by sampled-data output feedback controller with the same condition(3).

Theorem 1Under Assumption 1,the problem of global sampled-data output feedback stabilization for system(1)is solved.

ProofFor the convenience of the readers,we break up the proof into the following two parts.

2.1 Linear sampled-data output feedback controller design

We first introduce a change of the coordinates for system(1)

where 0<ε<1 is a constant to be determined later.

Under(4),system(1)can be written as

whereand

B=[0...0 1]T,C=[1 0...0].

Under Assumption 1,it can be verified that fori=1,...,n-1

Since only the outputy=z1is measurable at sampling points and the statesz2,...,znare not available,we will construct an observer to estimate the unmeasurable states of system(5).Motivated by[20],we design the following observer with continuous-time states over[tk,tk+1)and discrete-time outputz1(tk)and control input ν(tk)

whereai,i=1,...,nare coefficients of the Hurwitz polynomial

It is well-known that the continuous-time observer(8)with a sampler is equivalent to the following discretetime system:

where

Sincez2(t),...,zn(t)are not measurable,the sampleddata control law using the estimated states from(9)is constructed as

whereandki>0,i=1,...,nare coefficients of the Hurwitz polynomial

Substituting(10)into(9),it yields

whereM=F-GK.

In what follows,we use the observer(8)to analyze the stability of the closed-loop system.

2.2 Lyapunov analysis to determine the gain ε and period T

In this part,we present the Lyapunov design approach to choose the scaling gain ε and sampling periodT.Substituting the control law(10)into(5)and(8),the closed-loop system in the time interval[tk,tk+1)leads to

Due to the fact thatsystem(12)can be rewritten as

where

By construction,A-HCandA-BKare Hurwitz matrices.This,together with the following relation:

implies that A is also a Hurwitz matrix.Therefore,there exists a positive definite matrixP=PT∈R2nX2n>0 such that

Construct a Lyapunov functionwithA direct calculation yields

Next,we estimate the last three terms in the righthand side of(14).First,we estimate the termK(?z(tk)-?z(t)).Note that

By(8)and(10),for any τ∈[tk,t),

Moreover,from(17),it can be obtained that

By using(19),we have

Finally,we estimate the last term in the right-hand side of(14).According to(6),one has

By using(10),it is clear that

Substituting(22)and(17)into(21),it yields

With the help of(23),we have

Substituting(17),(20)and(24)into(14),one has

where

Therefore,we can choose the scaling gain ε and the sampling periodTas

wherec?>0 is a constant.

In what follows,we will verify the stability of the closed-loop system.With the help of(26)and(27),it is not difficult to obtain that

By a contradiction argument,we can obtain that there exits a time instantsuch thatV(Z(t1))>it is easy to prove from(26)that forAs a result,there is a time instantt2∈[tk,t1]such that

With this in mind,we can deduce from(26)that

which contradicts to the assumption˙V(Z(t2))>0.Therefore,we can conclude that(28)is true.

Based on(28),it follows from(26)that?t∈ [tk,tk+1),

Since ζ(tk)=1,we have

Hence,

From(27),it can be deduced that

under which we can obtain ρ<1 andV(Z(tk))converges to zero asktends to infinity.Consequently,by appropriately choosing the scaling gain ε and sampling periodT,system(1)can be rendered globally asymptotically stable via the sampled-data output feedback controller(10).

Remark 2A discrete-time state observer and a discrete-time control law are constructed in this paper with a sampling periodT,which is determined after we have fixed ε,in order to guarantee the closed-loop system globally asymptotically stable.

3 An illustrative example

In this section,we present an example to illustrate the effectiveness of the proposed scheme for feedforward nonlinear systems.

Consider the following system:

Obviously,it can be verified that the nonlinearities satisfy the condition(6)as|ln(1+(ε2z3)2+(ε3ν)2| ≤2ε2(|z3|+|ν|).

By Theorem 1,we can construct the sampled-data output feedback controller as follows:

By(27),we can choose ε=0.25,T=0.1,a1=16,a2=26,a3=11,k1=320,k2=152 andk3=22.Then it can be calculated from(9)and(10)thatN=[0.3359 0.5373 0.2260]Tand

The simulation results are shown in Figs.1 and 2 with initial condition[1 0.5 0 0.5-0.5 0].

Fig.1 State trajectories.

Fig.2 Control input.

4 Conclusions

In this paper,we investigate the problem of global stabilization for a class of feedforward nonlinear systems under sampled-data control.A sampled-data output feedback controller is constructed with an appropriate scaling gain and a sampling period which can guarantee the global stability for feedforward nonlinear systems.The proposed linear sampled-data controller can be easily implemented by computers.

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10 October 2013;revised 21 March 2014;accepted 24 March 2014

DOI10.1007/s11768-014-0160-8

?Corresponding author.

E-mail:jyzhai@seu.edu.cn.Tel.:+86-25-83792720.

This work was supported by the National Natural Science Foundation of China(Nos.61104068,61273119),Natural Science Foundation of Jiangsu Province(No.BK2010200),China Postdoctoral Science Foundation Founded Project(No.2012M511176),and the Fundamental Research Funds for the Central Universities(No.2242013R30006).

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

Zhihui WANGreceived his B.S.degree from Electrical and Electronic Engineering College,Yangzhou University in 2012.Currently,he is a M.S.candidate at the School of Automation,Southeast University,China.His research interests include nonlinear control and integration of switched systems.E-mail:836754527@qq.com.

Junyong ZHAIreceived his Ph.D.degree in Automatic Control from Southeast University in 2006.From September 2009 to September 2010,he was a postdoctoral research fellow at the University of Texas at San Antonio.He is currently an associate professor and doctoral advisor at the School of Automation,Southeast University.His current research interests include nonlinear systems control,stochastic time-delay systems,and multiple models switching control.E-mail:jyzhai@seu.edu.cn.

Shumin FEIreceived his Ph.D.degree from Beijing University of Aeronautics and Astronautics in 1995.From 1995 to 1997,he was a postdoctoral research fellow at Southeast University.Currently,he is a professor and doctoral advisor at Southeast University.He has published more than 100 journal papers.His research interests include nonlinear systems,stability theory of delayed system,complex systems.E-mail:smfei@seu.edu.cn.