Tao BIAN,Zhong-Ping JIANG

Control and Networks Lab,Department of Electrical and Computer Engineering,Tandon School of Engineering,New York University,5 Metrotech Center,Brooklyn,NY 11201,U.S.A.Received 18 November 2015;revised 9 December 2015;accepted 9 December 2015

New results in global stabilization for stochastic nonlinear systems

Tao BIAN?,Zhong-Ping JIANG

Control and Networks Lab,Department of Electrical and Computer Engineering,Tandon School of Engineering,New York University,5 Metrotech Center,Brooklyn,NY 11201,U.S.A.Received 18 November 2015;revised 9 December 2015;accepted 9 December 2015

This paper presents new results on the robust global stabilization and the gain assignment problems for stochastic nonlinear systems.Three stochastic nonlinear control design schemes are developed.Furthermore,a new stochastic gain assignment method is developed for a class of uncertain interconnected stochastic nonlinear systems.This method can be combined with the nonlinear small-gain theorem to design partial-state feedback controllers for stochastic nonlinear systems.Two numerical examples are given to illustrate the effectiveness of the proposed methodology.

Stochastic nonlinear systems,stochastic input-to-output stability,stabilization,gain assignment,small-gain

1 Introduction

Over the past few decades,global stabilization of deterministic nonlinear dynamical systems has attracted a lot of attention;see,e.g.,[1-10],and numerous references therein.In parallel,considerable efforts have also been made for the control of stochastic nonlinear systems[11-16].In particular,different methods based on variants of Sontag’s input-to-state stability(ISS)[17-20]were developed for stochastic nonlinear systems;see[21]for γ-input-to-state stability(γ-ISS);[13,22-25]for noise-to-state stability(NSS);and[15,26-28]for stochastic input-to-state stability(SISS).

Despite the above-mentioned developments,several issues still remain unresolved for stochastic nonlinear systems.One of such issues is how to generalize the existing global stabilization techniques[2-4]to stochastic systems.Although some efforts have been made[11,13-15,24,29-31]to solve this problem,these papers only considered state-dependent noise,and the control-dependent noise is overlooked.The main obstacle in conducting the control design in the presence of control-dependent noise is that using traditional control design methodologies has the potential to magnify the stochastic noise,and lead to instability for the closedloop system.Compared with the deterministic control design,extra conditions on the stochastic disturbanceare needed.

Another issue is how to generalize the deterministic ISS gain assignment results[8,32,33]to uncertain stochastic systems.Roughly speaking,the ISS gain assignment aims at finding a feedback control law under which the closed-loop system is ISS with a desired gain function.It is equivalent to pole placement for linear time-invariant systems.One major application of gain assignment is to design a robust partial-state feedback controller for systems subject to dynamic uncertainties.In these cases,if the ISS gain of the nominal system can besomehowfreelyassignedviaafeedbacklaw,thenthe ISS gains of the nominal system and the dynamic uncertainty will satisfy the small-gain condition[32],thereby guaranteeing the global stability of the closed-loop system.Indeed,gain assignment has played a crucial role in solving various nonlinear feedback control design problems[3,5,7,8].

This paper has at least two major contributions.First,we solve the global stabilization problem for stochastic nonlinear systems with control-dependent noise.Three nonlinear control design methodologies based on control Lyapunov function(CLF)[2]and Hamilton-Jacobi-Bellman(HJB)equation[6]are developed.Different from existing stochastic stabilization approaches[11,13-15,24],the obtained results can tackle the presence of control-dependent noise.Second,we extend the obtained stabilization technologies to conduct robust input-to-output gain assignment for a class of uncertain interconnected stochastic nonlinear systems.It can be shown that under some mild assumptions,given a nonlinear gain function,it is possible to develop a smooth feedback law under which the closed-loop system satisfies stochastic input-to-output stability(SIOS)with the desired gain function.Potential applications of the proposed stochastic gain assignment method include developing stochastic robust optimal control,stochastic output-feedback control,and decentralized control designs for large-scale stochastic networks,among other applications.

The remainder of this paper is organized as follows.In Section 2,we introduce three novel control design methods for stochastic nonlinear systems.In Section 3,the SIOS gain assignment is developed for a class of uncertain interconnected stochastic nonlinear systems.Two simulation examples are presented in Section 4 to illustrate our results.Finally,the conclusion is drawn in Section 5.

NotationsThroughout this paper,R denotes the set of real numbers.R+denotes the set of nonnegative real numbers.Indenotes the identity matrix of dimensionn.|·|denotes the Euclidean norm for vectors,or the induced matrix norm for matrices.|·|Fdenotes the Frobenius norm for matrices.A real-valued functionfis of classwithka nonnegative integer andQ?Rn,iffis continuous whenk=0,orfisk-times continuously differentiable onQwhenk≥1.A functionf:Q→R+,whereand 0∈Q,is called positive definite,iff(x)＞0 for allx∈Qandf(0)=0.fis called proper,ifQ=and∞.A function γ :is of classif it is continuous,strictly increasing,and γ(0)=0;it is of classif,in addition,A function β :is of classif β(·,t)is of classfor any givent,and β(s,·)is decreasing and vanishes at the infinity for any givens.Given a measurable functionu:=andutis the truncation ofuon[0,t]satisfyingut(s)=u(s)for alls∈[0,t],andut(s)=0 otherwise.Denote bywaq-dimension standard Wiener process,andas the σ-field generated byw(s),0 ≤s≤t.L denotes the differential generator[34].

2 Stabilization of stochastic nonlinear systems with control-dependent noise

2.1 Basic notions for deterministic systems

In this section,we introduce some important concepts of nonlinear system theory.These concepts play an important role in deriving the main results in this paper.

Control Lyapunov functionsControl Lyapunov function(CLF)waspreviouslyproposedin[1,2]totackle thenonlinearstabilizationforgeneralnonlinearsystems.Consider a deterministic affine nonlinear system

wherex∈Rnis the state;u:R+→Rmis a measurable locally essentially bounded input signal;f:Rn→Rnandg:Rn→ Rn×mare locally Lipschitz functions.f(0)=0.

Now,we give the definition of CLF.

De fi nition 1A positive definite and proper functionis said to be a CLF for system(1),if for all

where for allx∈Rn,

It has been shown in[2]that(2)is equivalent to

A nice feature of CLF is that the existence of a CLF is a necessary and sufficient condition for the global stabilizability of system(1).Based on a given CLF,a universal formula of controller design was proposed in[2].

Proposition 1Consider system(1).Given a CLFV,one can construct the following feedback law μ∈

whereb(x)=(LgV(x))T,such that system(1)underu=μ(x)is globally asymptotically stable(GAS)at the origin.

Note that μ is not necessarily continuous at the origin.To tackle this problem,we can assume thatVpossesses small control property[2]:For each ε＞0,there is a δ ＞ 0 such that,ifx≠ 0 satisfies|x|＜ δ,then there is someuwith|u|＜ ε such that

Input-to-state stabilityAnother important topic in nonlinear control theory is the disturbance attenuation problem.A key concept in this field is ISS[17].

De fi nition 2System(1)is said to be input-to-state stable,if there existand ρ ∈,such that for allt≥ 0,x(0)∈Rn,

ρ is usually referred to as the ISS gain of system(1).

An equivalent definition in terms of ISS-Lyapunov function is given below.The proof of the following lemma can be found in[19].

Lemma 1System(1)is input-to-state stable,if and onlyifthereexistsapositivedefiniteandproperfunctionsuch that for allx∈Rnandu∈Rm,

Vis known as an ISS-Lyapunov function.

In the sequel,we introduce three control design methodologies for stochastic nonlinear systems.

2.2 CLF approach

Consider the following stochastic system:

wherex∈Rnis the state;u:adapted stochastic process representing the control input;f,gare as in system(1),andg0:is continuous and satisfiesg0(0,0)=0.

First,an assumption is made on the diffusion term.Assumption 1There exist continuous functions κi:Rn→R+,i=0,1,such that for allx∈Rn,u∈Rm,

Moreover,κ0(0)= κ1(0)=0.

Next,assume the nominal system of(3)(without the diffusion term)is stabilizable.

Assumption 2There exista positive definite functiona positive definite and proper functionand a nonnegative functionsuch that for allx∈ Rn,

Theorem 1Consider(3).Under Assumptions 1 and 2,if

where κ2(|x|) ≥ |μ(x)|,for allx∈ Rn,then,system(3)underu=μ(x)is GAS at the origin with probability one.

ProofFor allx∈Rn,

By(4),we know system(3)underu=μ(x)is GAS at the origin with probability one.This completes the proof. □

Remark 1Note from Theorem 1 that when the stochastic noise is control-dependent,the stability is guaranteed either whenuhas a small influence on the stochastic noise,or a low-gain controller is used in the control design.

Remark 2It is worth noticing that(4)can be regarded as a generalization of the small-gain condition[32]for deterministic nonlinear systems.

Example 1Consider a scalar system

ObviouslyAssumption2holdsforallandThen,as long aswe can chooseu=-|x|/(2ε2),such that the closed-loop system is GAS at the origin with probability one.

2.3 HJB equation approach

Consider the following stochastic system:

wherex,u,fandgfollow the definitions in Section 2.2;andg0i:Rn→ Rn×m,i=1,2,···,q,are continuous.

Theorem 2Consider system(5).If there exist a positive definite functionand a continuousR:Rn→ Rm×m,withR(x)=RT(x)＞ 0 for allx∈ Rn,such that the following HJB equation

admits a positive definite and proper solutionV∈satisfying

Then,there exists a feedback law μ∈such that system(5)underu=μ(x)is GAS at the origin with probability one.

ProofDefine

Thus,system(5)underu=μ(x)is GAS at the origin with probability one.This completes the proof. □

Remark 3It is of interest to note that Theorem 2 recovers[35,Theorem 5]as a special case.

2.4 Backstepping

In this section,let us consider the following single input interconnected stochastic nonlinear systems:

Remark 4The model(6)and(7)is in a more general form than the strict-feedback model investigated in[13,Chapter 3.2]and[15].

Before proceeding,we denote

for allx1∈Rnandx2∈R.

Assumption 3There exist a smooth CLFV:Rn→R+,a smooth feedback law μ0:Rn→ R,κ :R+→ R+with κ(0)=0,and a positive definite function α ∈such that

for allx1∈Rn.

Assumption 4There exist=0,1,2,such that

Theorem 3Consider system(6)and(7).Under Assumptions 3 and 4,iff,g1,andg2are smooth,and

for allx1∈Rn,then there exists a smooth feedback law μ:Rn×R→R,such that system(6)and(7)withu=μ(x1,x2)is GAS at the origin with probability one.

ProofDenotez=x2-μ0(x1).Then,by It?o’s lemma[36,Page 16],

Then,Assumption 3 holds withV=/2 and μ0(x1)=-x1/2.By Theorem 3,we can design a state-feedback controlleru,such that the system is GAS at the origin with probability one.

Remark 5Theorem 3 requires the precise knowledge of system(6)and(7)and the seemingly restrictive condition(8).These two drawbacks are relaxed in the next section for a subclass of interconnected stochastic systems.

3 Gain assignment for a class of interconnected stochastic systems

Consider the following second-order stochastic systems:

whereis the state;u:R+→R andv:processes representing the control input and the disturbance input,respectively;f1:R2×Rm→R,f2:R2×R×Rm→R,g1:R×Rm→Rq,andg2:R2×Rm→Rqare uncertain locally Lipschitz functions.f1(0,0)=f2(0,0,0)=0,g1(0,0)=g2(0,0)=0.

First,following[26,Definition 2.1],we give the concept of SIOS.

De fi nition 3The uncontrolled system(10)-(12)withu≡0 is SIOS,if for any 0＜?≤1,there existsuch that for allt≥ 0 and ξ∈R2,

wherePξ,0{A}denotes the probability of an eventA∈,givenx(0)=

ρ is known as the SIOS gain of system(10)-(12).

In this section,our aim is to solve the following control problem:

Problem 1(Gain assignment)Consider system(10)-(12).For a given ρ ∈find a static statefeedback controlleru= μ(x),where μ,such that for all ?∈(0,1],the closed-loop solutions satisfy(13)for somedepending on ?.

To address Problem 1,the following assumption is imposed on system(10)and(11).

Assumption 5There exist εi∈ [0,1),i=1,2,and smooth functions κi:R+→ R+,i=1,2,3,4,such that for allx∈R2,u∈R,andv∈Rm,

Before giving the main result,a lemma is given.The proof of Lemma 2 can be found in[26].

Lemma 2Consider the uncontrolled system(10)and(11)withu≡ 0.Suppose there exist αii= 1,2,and positive definite functionssuch that for allx∈R2,

Then,for any ?∈ (0,1],t≥ 0,and ξ ∈ R2,

Theorem 4Consider system(10)-(12).For anywith ρ-1(|x1|)/|x1|locally bounded by a smooth function at the origin,if Assumption 5 holds,then there exists a feedback lawsuch that for system(10)-(12)withu=μ(x),(13)holds for any 0＜?≤1 with the given ρ and some β?∈

ProofThe proof is inspired by[13,Chapter 3.2].First,denotez=x2- μ0(x1)andu=-zχ2(x),where μ0(x1)=-x1χ1(x1),and χ1:R → R+and χ2:R2→R+are smooth functions given later.Then,by It?o’s lemma[36,Page 16],

Since ρ-1(|x1|)/|x1|is locally bounded by a smooth function at the origin,there exists a strictly increas-ingsuch thatall x1∈ R.For anyone has from Assumption5 and Young’s inequality[13,Corollary A.3]that

Now,we can choose χ1and χ2,such that

By Lemma 2,we know the system composed by(10)and(16)satisfies(15).Thus,(13)holds with the given ρ and someThis completes the proof. □

Remark 6It is worth noticing that we can consider higher-ordersystemsbyapplyingthebacksteppingtechnique in the proof of Theorem 4 repeatedly.

Example 3Consider a scalar system

Obviously Assumption 5 holds.Then,for any ρ ∈satisfying the conditions in Theorem 4,we can chooseu=-x(1+ γ(x)+3/2x2+3/2γ2(x)),where γ(x)=ρ-1(|x|)/|x|,such that(13)holds for any ?∈ (0,1]with some

4 Numerical results

4.1 Stochastic IOS gain assignment for uncertain system

To illustrate the obtained results in Section 3,let us consider an elementary example of a second-order nonlinear system:

wherewis the standard Wiener process and Δ is an unknown function satisfying

Then,all the conditions in Theorem 4 hold.Note that[13,Example 3.6.3]can be considered as a special case of system(18)and(19).

Suppose the gain function in(13)is given as ρ(s)=s1/3,for alls≥0.By Theorem 4,the controller can be chosen as

The plots of the closed-loop states corresponding tox(0)=[0.2-1.2]Tare shown in Fig.1.

Fig.1 Plots of the trajectories x1and x2.

4.2 Single-joint human arm movement

In this example,a numerical simulation of the singlejoint human arm movement is given to illustrate the obtained results.

The dynamic model of the single-joint human arm movement is given below[37]:

wherem=1.65kg is the mass of segment,I=0.0779 kg·m2is the inertia,g=9.81m/s2is the gravitational constant,l=0.179m is the distance of the center of mass from the joint,θ is the joint angular position,Tmis the input to the muscle from the motoneurons,andndenotes the inputs from the neural integrator satisfying

wherecudwrepresentsthesignal-dependentnoise[39];c=0.05.Note that(20)is regarded as the dynamic uncertainty.

Based on our design scheme,the controller is chosen as

The plots of closed-loop trajectories under the proposed controller given above are shown in Fig.2.

Fig.2 Simulation of the single-joint human arm movement.

5 Conclusions

Thispaperhasinvestigatedglobalstabilizationandrobust gain assignment problems for stochastic nonlinear systems.Three nonlinear control design methodologies based on CLF and HJB equation have been developed.Moreover,a robust SIOS gain assignment technique has been proposed for a class of uncertain interconnected stochastic nonlinear systems.The effectiveness of the obtained results is demonstrated by two simulation examples.Future extensions of the proposed results include data-driven nonlinear control[40-43],event-triggered control[44,45],and decentralized control[8,24,33].

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DOI10.1007/s11768-016-5117-7

?Corresponding author.

E-mail:tbian@nyu.edu.Tel.:+1 7182603779.

This work was partially supported by the National Science Foundation(Nos.ECCS-1230040,ECCS-1501044).

the B.Eng.degree in Automation from Huazhong University of Science and Technology,Wuhan,China,in 2012,and the M.Sc.degree in Electrical Engineering from New York University,NY,in 2014,where he is currently working toward the Ph.D.degree.His research interests include approximate/adaptive dynamic programming,nonlinearcontrol,andstochastic control systems..E-mail:tbian@nyu.edu.

Zhong-Ping JIANGreceived the B.Sc.degree in Mathematics from the University of Wuhan,Wuhan,China,in 1988,the M.Sc.degree in Statistics from the Universite de Paris-sud,France,in1989,andthePh.D.degree in Automatic Control and Mathematics fromtheEcoledesMinesdeParis,France,in 1993.Currently he is a Professor of Electrical and Computer Engineering at New York University Tandon School of Engineering(formerly called Polytechnic University).His main research interests include stability theory,the theory of robust and adaptive nonlinear control,and their applications to underactuated mechanical systems,congestion control,wireless

networks,multi-agent systems and systems physiology.Dr.Jiang has served as a Subject Editor for the International Journal of Robust and Nonlinear Control,and as an Associate Editor for Systems&Control Letters,IEEE Transactions on Automatic Control and European Journal of Control.Dr.Jiang is a recipient of the prestigious Queen Elizabeth II Fellowship Award from the Australian Research Council,the CAREER Award from the U.S.National Science Foundation,and the Young Investigator Award from the National Natural Science Foundation of China.He(together with coauthor Yuan Wang)received the Best Theoretic Paper Award at the 2008 World Congress on Intelligent Control and Automation,June 2008,for the paper“A Generalization of the Nonlinear Small-Gain Theorem for Large-Scale Complex Systems”.Dr.Jiang is a Fellow of the IEEE and a Fellow of IFAC.E-mail:zjiang@nyu.edu.