LI Guifangand Ye-Hwa CHEN

1.College of Civil Aviation,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China;

2.George W.Woodruff School of Mechanical Engineering,Georgia Institute of Technology,Georgia 30332,USA

1.Introduction

The mathematical model of systems always contains uncertainty.While no information on the uncertainty other than its possible bound is assumed for the design,it is very essential to categorize the structure of system in order to capture the effect of the uncertainty on the system performance.In the past,numerous controls were constructed based on a structural condition on the system,namely,the matched condition.That means that the uncertainty must be within the range space of the nominal input matrix.Under the framework of deterministic systems,there have been some results on the stability or boundedness for uncertain systems if the matched conditions are satisfied.In[1]and[2],it were shown that if the so-called matched conditions on the uncertainty are satisfied,one can design appropriate controls such that certain stability can be guaranteed for uncertain linear systems.In[3],Gutman designed discontinuous min-max controllers to asymptotically stabilize nonlinear dynamic systems with matched uncertainties.And then,Corless and Leitmann[4]developed continuous state feedback controllers for guaranteeing uniform boundedness of matched uncertain system trajectories.Moreover,Barmish[5]introduced the concept of practical stability and proposed stabilizing controllers for systems with matched uncertainties via the Lyapunov stability method.Differnent from[5],Kravaris[6]used the differential geometric approach to stabilize a class of matched uncertain dynamic systems.However,the systems they considered all were deterministic systems.

However,many real systems are stochastic,so the analysis and design of stochastic nonlinear control systems have received considerable attention in the past years.Some fundamental notations[7],stability theory[8]as well as design tools[9–11]are extended to stochastic cases from the deterministic nonlinear systems.Furthermore,the robust and adaptive control problems are discussed when there exist uncertainties in stochastic nonlinear systems.The back stepping technique provides a new way to solve many control problems for the strict-feedback stochastic systems[12–14]and the non-strict-feedback stochastic systems[15,16].On the basis of it,the dynamic surface control approach[17,18]makes some improvement which overcomes the “parameter explosion”phenomenon,and has attracted more and more attention.Recently,fuzzy theory[19–22]and neural network method[23–25]have been widely applied to deal with the control problems for stochastic nonlinear systems with unknown nonlinearity and/or un modeled uncertainty.The finite-time control problem for stochastic nonlinear systems was also discussed in[26–28].However,we expect to acquire the effect of uncertainty on stochastic systems performance and design more simple controller under matched conditions.

This paper addresses the global boundedness problem for a class of stochastic nonlinear systems with matched conditions.Under the assumption that its nominal system is globally bounded in probability,the controller can be de-rived through the gradient method to guarantee responses of stochastic nonlinear systems globally bounded in probability.The controller will degrade to the linear one when the systems are linear.

2.Notations and preliminary

Consider the stochastic nonlinear system

where x∈Rnis the state,w is an r-dimensional independent standard Wiener process,f(·)∶Rn× R → Rnand g(·)∶Rn× R → Rn×rare locally Lipschitz.

define the in finitesimal generator of V along(1)by

definition 2[29] Consider the system(1)with f(0,t)=0,g(0,t)=0.The equilibrium x=0 is globally stable in probability if for any ε＞ 0,there exists a class of K function γ(·)such that P{|x(t)| ＜ γ|x0|}?1- ε,?t?0,x0∈ Rn{0}.

The following theorem givens sufficient conditions on the boundedness and stability properties for the system(1),which is based on the stochastic Lyapunov stability theory.

Theorem 1[30] Consider the stochastic systems(1)and assume that f(x,t),g(x,t)are C1in their arguments and f(0,t),g(0,t)are bounded uniformly in t.If there exists functions V(x,t) ∈ C2,1,μ1(·),μ2(·) ∈ K∞,constants c1＞0,c2?0,and a nonnegative function W(x,t),such that

then for(1)there exists an almost surely unique solution on[0,∞);the solution process is globally bounded in probability,when W(x,t)?cV(x,t)for some constant c＞0;when c2=0,f(0,t)≡0,g(0,t)≡0 and W(x,t)=W(x)are continuous,the equilibrium x=0 is globally stable in probability and the solution x(t)satisfies

3.Uncertain stochastic systems

We consider the uncertain stochastic systems described by the state equation

where u∈Rmis the control,q∈Q?Rpis the uncertain parameter and the set Q is compact.f(x,t),Δf(x,q,t),B(x,t)and ΔB(x,q,t)are matrices of appropriate dimensions which depend on the structure of the system.

The objective of this paper is to design an appropriate controlleru(t),such that responses of the closed-loopsystems are globally bounded in probability.

We assume the following matched conditions in order to find out the effect of uncertainty on system performance.

Assumption 1There are continuous vector function h(·)∶Rn×Rp×R → Rmand E(·)∶Rn×Rp×R →Rm×msuch that

for all x∈Rn,q∈Q and t∈R.

Assumption 2The mappings f(·) ∶Rn× R →Rn,B(·)∶Rn×R → Rn×mandg(·)∶Rn×R → Rn×rare continuous.

Assumption 2 can guarantee the existence of solutions of the state equation(5).

We introduce the following assumption.

Assumption 3There exist a C2,1function V(·) ∶Rn×R→[0,∞)and strictly increasing continuous functions μ1(·),μ2(·) ∈ K∞,constants k1＞ 0,k2?0,such that

for all pairs(x,t)∈Rn×R.

This assumption,in effect,asserts the existence of a Lyapunov function for the nominal system such that system(9)is globally bounded in probability.

4.Controller design and bounded in probability

Firstly,we select functions Δ1(·)and Δ2(·) ∶Rn× R →R satisfying

One simply selects any continuous function μ(·) ∶Rn×R→[0,∞)satisfying

where ε1,ε2are any(designer chosen)non-negative constants.

(i)ε1＜ k1;

We propose the control

In fact,(14)and(15)describe a class of controllers yielding global boundedness.Sometimes it includes linear controllers.

Theorem 2Subject to Assumptions 1–3,the stochastic nonlinear uncertain systems(5),(14)and(15)are globally bounded in probability.

ProofFor any given uncertainty q∈Q,the in finitesimal generator for the systems(5)with the feed back control(15)is given by

where

By Assumption 1,it becomes

Denote φ(·)∶Rn× R → Rmas

By using of the definition of Δ1(·)and Δ2(·),we have

Next,we consider two cases.

(i)If Δ1(x,t)=0,(20)has the following property:

(ii)If Δ1(x,t)/=0,it follows from(14)that μ(x,t) ＞0.Moreover,in view of(14),we have

Combining(i)and(ii),the closed-loop systems(5),(14)and(15)are globally bounded in probability according to Theorem 1. ?

5.Specialization to linear systems

We consider the special case when

where A,ΔA(q),B and ΔB(q)are matrices of appropriate dimensions and q∈Q.The uncertainties follows that

To obtain a Lyapunov function for the nominal systems,we simply select an n×n positive-definite symmetric matrix Q,which satisfies

for P which is positive-definite.Consider the Lyapunov candidate function

Along the solution of(23),we have

where ε1=min{λi(Q)/λi(P)},i=1,2,...,dim(P),

Using the notations above,we define

Then,in agreement with(12)and(13),we may take

One can select μ(·)such that

This implies that the control(15)is a linear time-invariant feedback,that is,

Theorem 3If A is Hurwitz,the stochastic linear uncertain systems(23),(32)and(33)are globally bounded in probability.

RemarkWhen the systems(5)are linear,the controller can be realized as a linear feedback.

6.Illustrative examples

Example 1Consider the stochastic nonlinear systems

where a＞0 is an uncertain constant,and|a|≤a.In order to satisfy Assumption 1 and Assumption 3,one can assure the nominal system bounded.Hence,we propose a controller of the form

where b and c are positive constants.Substitution of(36)into(34)and(35)yields the state equation

where

The nominal systems of(37)is

where

Firstly,given Q,we get the solution P from

Furthermore,one can select the Lyapunovfunctionas V=xTPx.Along the solution of(39),the in finitesimal generator of V is shown as

And

so we have

And then,one can choose

Finally,the controller is specified by

The parameters in(36)are b=0.05,c=0.2.

And the controller is given by

Under the designed controller(48),the systems(34),(35)are globally bounded in probability.

Fig.1 shows that responses of the systems(34),(35)are globally bounded under control(48).Comparing Fig.1 with Fig.2,we can see the control law(48)has better control effect and it is also bounded in Fig.3.

Fig.1 Responses of the closed-loop system of(34)(35)and(48)

Fig.2 Uncontrolled states(34)and(35)

Fig.3 Control law(48)

Example 2Consider the stochastic linear systems

where q is an unknown constant and q∈[-0.5,0.5].We have D(q)=[1 q],E(q)=q andρD=1.12,ρE=0.50.

And the control law is

Figs.4–6 show that responses of the closed-loop systems(49),(51)are globally bounded and the control law(51)is also bounded.

Fig.4 Responses of the closed-loop system(49)and(51)

Fig.5 Uncontrolled state(49)

Fig.6 Control law(51)

7.Conclusions

This paper addresses the global boundedness in probability for a class of stochastic nonlinear systems with matched uncertainty.When the systems degrade to linear systems,and the controller becomes linear.

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