state-feedback stabilization for a class of more general high order stochastic nonholonomic systems

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2011; 25:687–706Published online 4 March 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/acs.1233

State-feedback stabilization for a class of more general high orderstochastic nonholonomic systems

Yan Zhao1, Jiangbo Yu1 and Yuqiang Wu2,∗,†

1School of Automation, Southeast University, Nanjing, Jiangsu 210096, People’s Republic of China2Institute of Automation, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China

SUMMARY

This paper studies the problem of state-feedback stabilization control for a class of high order stochasticnonholonomic systems with disturbed virtual control directions and more general nonlinear drifts. Byusing the backstepping approach, we develop a recursive controller design procedure in the stochasticsetting. To get around the stabilization burden associated with nonholonomic systems, a switching controlstrategy is exploited in this procedure. The tuning function technique is applied in the design to avoidthe disadvantage of overparameterization. It is shown that, under some conditions, the designed controllercould ensure that the closed-loop system is almost asymptotically stabilized in probability. It is notedthat the obtained conclusion can be extended to multi-input systems. A simulation example is providedto illustrate the effectiveness of the proposed approach. Copyright � 2011 John Wiley & Sons, Ltd.

Received 29 June 2010; Revised 28 October 2010; Accepted 23 January 2011

KEY WORDS: state-feedback; high order stochastic nonholonomic systems; backstepping; tuning functiondesign; switching control strategy

1. INTRODUCTION

The research on nonholonomic control systems shows high significance both in theory and appli-cation, since it represents a wide class of mechanical systems with nonholonomic (nonintegrable)constraints. The wheeled mobile robot, knife edge, and rolling disk, for instance, are all repre-sentative examples of such systems [1]. From Brockett’s necessary conditions for stability [2], itis known that a nonholonomic system is not stabilizable by stationary continuous state-feedback.During the past decades, a number of researchers have been attracted to seek for novel approachesto overcome this obstruction, and many fruitful results have been generated, see [3–13]. In liter-atures, three methods are adopted for stabilization of nonholonomic systems, i.e. discontinuoustime-invariant stabilization [4–7], smooth time-varying stabilization [8, 9], and hybrid stabilization[10]. With these methods, the stabilization problem for several classes of nonholonomic systemsis solvable. However, this problem for high order stochastic nonholonomic systems has not beenreceived much attention.

It is known that a nonholonomic system could be transformed into a nonlinear system throughstate-input scaling in [13]. With the help of this tool, the nonholonomic control problem canbe investigated from the following two aspects. One is the research of nonholonomic systems.There has been much advancement in this field. For example, a class of high order nonholonomic

∗Correspondence to: Yuqiang Wu, Institute of Automation, Qufu Normal University, Qufu, People’s Republic ofChina.

†E-mail: [email protected]

Copyright � 2011 John Wiley & Sons, Ltd.

688 Y. ZHAO, J. YU AND Y. WU

systems in power chained form is considered in [14]. The obtained switching controller rendersthe closed-loop systems almost asymptotically stabilized. Also, in the deterministic setting, thisresult is recently extended to the systems with nonlinear drifts in [15]. With unknown covariancestochastic disturbances, such class of nonholonomic systems in chained form is investigated in[16]. The other is the investigation of the nonlinear systems. In [17], for a class of systems whosenonlinearities fi (·) rely on (x1, . . . , xi+1), a continuous controller is designed to achieve the globalstrong stabilization. By strengthening conditions, it is extended to the stochastic case in [18].However, only part of the considered nonlinearities fi (·) are dependent on (x1, . . . , xi+1). Naturally,it is of interest to generalize this result to the nonholonomic systems with more general nonlineardrifts in the stochastic setting.

This problem will be further investigated here. In this paper, the system under considerationis more general than those investigated in [14–16]. The nonlinear functions fi are all dependenton (x1, . . . , xi+1). Moreover, the system has the disturbed virtual control directions but a prioriknowledge of their sign and boundedness. Our work has three aspects. First, we transform thenonholonomic system into a nonlinear one. Second, the backstepping technique is applied to designa smooth controller for the transformed system. Third, we present a switching control strategyfor the original system, which guarantees that the closed-loop system is almost asymptoticallystabilized at the origin in probability and the state is globally asymptotically regulated to zero inprobability.

2. PROBLEM FORMULATION

In this paper, we focus our attention on the following class of high order nonholonomic systems:

dx0 = d0(t)u p00 dt,

dxi = di (t)xpii+1uqi

0 dt + fi1(x0, x1, . . . , xi+1)dt + f Ti2(x0, x1, . . . , xi+1)�(t)d�,

dxn = dn(t)u pn dt + fn,1(x0, x,u)dt + f Tn,2(x0, x,u)�(t)d�, i =1, . . . ,n−1,

(1)

where (x0, xT)T = (x0, x1, . . . , xn)T ∈Rn+1 is state, u0,u ∈R are control inputs, di (t)(i =1, . . . ,n)are unknown functions, fi1 :Ri+2 →R and fi2 :Ri+2 →Rr are smooth functions satisfyingfij(0,0, . . . ,0)=0(i =1, . . . ,n; j =1,2). Assume that pi�1(i =0,1, . . . ,n) are odd integers andqk(k =1, . . . ,n−1) are positive integers; � is an r -dimensional independent standard Winerprocess with incremental covariance �(t)�(t)T dt , i.e. E{d�d�T}=�(t)�(t)T dt , where �(t) is abounded function taking values in the set of nonnegative definite matrices for each t .

In the remainder of this paper, we will first design a state-feedback controller u for system(1) such that, given any initial state (x0(0), x(0))∈�0 ={(x0, x)|x0 �=0}, the closed-loop systemis almost asymptotically stabilized in probability [4, 5, 16]. Then, a switching control strategy isproposed, which guarantees that all the signals are bounded in probability for all initial conditions(x0(0), x(0))∈Rn+1.

In order to achieve the above control objective, throughout the paper, we make the followingassumptions regarding system (1).

Assumption 1For i =1, . . . ,n, there exist known smooth nonnegative functions �ij, �ij satisfying

fi1(x0, xi+1)�pi −1∑j=0

|xi+1| j (|x1|pi − j +|x2|pi − j +·· ·+|xi |pi − j )�ij(x0, xi ), (2)

fn1(x0, x,u)�pn−1∑j=0

|u| j (|x1|pn− j +|x2|pn− j +·· ·+|xn|pn− j )�nj (x0, x), (3)

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2011; 25:687–706DOI: 10.1002/acs

STATE-FEEDBACK STABILIZATION FOR STOCHASTIC NONHOLONOMIC SYSTEMS 689

fi2(x0, xi+1) �[(pi −1)/3]∑

j=0|xi+1| j (|x1|((pi +1)/2)− j+|x2|((pi +1)/2)− j+·· ·+|xi|((pi +1)/2)− j )�ij(x0, xi ),

(4)

fn2(x0, x,u) �[(pn−1)/3]∑

j=0|u| j (|x1|((pn+1)/2)− j+|x2|((pn+1)/2)− j+·· ·+|xn|((pn+1)/2)− j )�nj (x0, x),

(5)

where [(pi −1)/3] represents the integral part of (pi −1)/3.

Assumption 2The following inequality holds: p0�p1� · · ·�pn .

Assumption 3For any t�0, there exist known positive constants � and � such that

��di (t)��, i =0,1, . . . ,n.

Remark 1In the recent work [18], the gain functions of the system noise fi2(i =1, . . . ,n) only depend onx1, . . . , xi . Moreover, fn1 is independent of u. That is to say, in [18], the nonnegative functions�nk and �ij (k =1,2, . . . , pn −1; i =1,2, . . . ,n; j =1,2, . . . , [(pi −1)/3]) in (3)–(5) are all equal tozero with �n0>0, �l0>0(l =1,2, . . . ,n). Clearly, both the state-dependent uncertainties and input-dependent uncertainties are involved in our case. This assumption here is less restrictive and allowsfor a much broader class of systems. For example, if fn2 does not depend on u, i.e. the power ofu is zero, (5) becomes

fn2(x0, x)�(|x1|(pn+1)/2 +|x2|(pn+1)/2 +·· ·+|xn|(pn+1)/2)�n(x0, x),

where �n(x0, x) is a smooth nonnegative function. And the case of fij(i =1, . . . ,n; j =1,2)independent of xi+1 is described in Corollary 1 of Section 4.

Remark 2Now let us see the connection of system (1) to a real-world example. Consider the mobile robotof tricycle type with parametric uncertainties

xc = p∗1v cos�,

yc = p∗1v sin�,

� = p∗2�,

(6)

where p∗1 and p∗

2 are unknown parameters taking values in a known interval [pmin, pmax] with0<pmin<pmax<∞, v and � are two control inputs which are viewed as the linear velocity andthe angular velocity of the system [19].

Here we assume the forward velocity v is subject to some stochastic disturbances and the angularvelocity � is uncertain. Since v is a state-feedback law, it is a function of xc, yc,� and can beexpressed as

v(xc, yc,�)=v1(xc, yc,�)+v2(xc, yc,�)B(t),

where B(t) is the so-called white noise, see pages 1–2 of [20]. Then system (6) is transformed into

dxc = p∗1v1 cos�dt + p∗

1v2 cos�dB,

dyc = p∗1v1 sin�dt + p∗

1v2 sin�dB,

� = p∗2�.

(7)



For system (7), introduce the following transformation:

x0 =�, u0 =�, u =v1

x1 = xc sin�− yc cos�

x2 = xc cos�+ yc sin�

and we obtain

dx0 = p∗2u0 dt,

dx1 = p∗2 x2u0 dt,

dx2 = p∗1u dt − p∗

2 x1u0 dt + p∗1v2 dB.

(8)

In view of the above transformation, � is a function of xc, yc, � as well as x0, x1, x2. As thereexist smooth functions ϑ1(x0, x1, x2) and ϑ2(x0, x1, x2) such that

�2(x0, x1, x2)= x1ϑ1(x0, x1, x2)+x2ϑ2(x0, x1, x2),

system (8) is a special case of system (1) with p0 = p1 = p2 =q1 =1, f11 = f12 =0, f21 =−p∗2 x1u0,

and f22 = x1ϑ1(x0, x1, x2)+x2ϑ2(x0, x1, x2).Next we introduce two lemmas which are crucial in establishing the main result of this paper.

Lemma 1 (Young’s inequality)If the constants p,q>1 and satisfy (p−1)(q −1)=1, then for all ε>0 and (x, y)∈R2, we have

xy�ε p

p|x |p + 1

qεq|y|q .

Lemma 2For stochastic nonlinear system dx = f (x)dt +gT(x)d�, where x ∈Rn , f (x) and g(x) are locallyLipschitz functions, if there exist C2 function V :Rn →R+, �1, �2 ∈K∞, such that

�1(‖x‖)�V (x)��2(‖x‖),

LV (x)= �V

�xf (x)+ 1

2T r

{g(x)

�2V

�x2gT(x)

}�−W (x),

where W (x) is a nonnegative continuous function [21]. Then for any x(0)∈Rn ,1. There exists an almost surely unique solution on [0,∞).2. The equilibrium x =0 is globally stable in probability, and the solution x(t) satisfies

P{

limt→∞ W (x(t))=0

}=1.

3. INDUCTIVE DESIGN PROCEDURE

In this section, we will consider the system (1) under the condition of x0(t0) �=0 and the case ofx0(t0)=0 will be discussed in the next section. Following this line, we will first show how to designa state-feedback control law to stabilize all the signals of the closed-loop system in probabilitywhen x0(t0) �=0. The design procedure is divided into two separate stages. First, we choose u0such that the state x0 is asymptotically stable. Then, the second control input u is given whichrenders the other signals in system (1) be bounded in probability.

Considering x0(t0) �=0, one can take u0 as follows:

u0 =−�0x0, (9)

where �0>0 is a constant. It will be shown that with the control law u0 in the form of (9), thestate x0 will not be equal to zero for all t�0, although it is regulated to zero as t tends to ∞.



Lemma 3For any initial condition (t0, x0(t0)) with t0�0, if x0(t0) �=0 and u0 is taken as in (9), then thesolution x0(t; t0, x0(t0)) is asymptotically stable and will not be zero for t�t0.

ProofThe conclusion can be testified by using the Lyapunov function V0(x0)= 1

2 x20 . In fact, it is easy to

verify that the time derivative of V0(x0) along the x0-subsystem of (1) satisfies

V0 = x0 x0 =d0(t)x0u p00 .

Noting that p0 is an odd integer, a direct substitution leads to

V0 =−�p00 d0(t)x p0+1

0 �−��p00 x p0+1

0 �0. (10)

As a consequence, the x0-subsystem is asymptotically stable. In addition, from (9), we have

x0 =−d0(t)�p00 x p0

0 .

If p0 =1, then the solution x0(t)= x0(t0)e−�0

∫ tt0

d0()d, which indicates that x0(t) �=0.

If p0>1, according to Assumption 3, there holds

x1−p00 (t) = x1−p0

0 (t0)+�p00 (p0 −1)

∫ t

t0d0()d

� x1−p00 (t0)+�p0

0 (p0 −1)�(t − t0).

Since p0 is an odd integer, we have

x0(t)�(x1−p00 (t0)+�p0

0 (p0 −1)�(t − t0))1/(1−p0)

or

x0(t)�−(x1−p00 (t0)+�p0

0 (p0 −1)�(t − t0))1/(1−p0).

This shows that, if x0(t0) �=0, x0(t) does not become zero when t�t0. The proof is completed. �

From Lemma 3, it can be seen that the state x0(t) �=0, ∀t�t0 under the control law u0 in (9).However, in the limit case, x0 will converge to the origin, which will cause serious trouble incontrolling the x-subsystem via the control input u. This difficulty can be well addressed byutilizing the following discontinuous input-state scaling transformation:

z1 = x1

ur10

, z2 = x2

ur20

, . . . , zn = xn

urn0

(11)

with the positive integers ri =qi + piri+1, rn =0, i =1, . . . ,n−1. Clearly, there holds

r1�r2� · · ·�rn−1�rn. (12)

In terms of (9), the transformation (11) has no definition at x0(t)=0. However, from Lemma 3 weknow that it makes sense when x0(t0) �=0.

Given the new z-coordinate, the x-subsystem of (1) can be rewritten into

dzi = di (t)zpii+1 dt +

(fi1(x0, xi+1)

uri0

−ri d0(t)ziu p0

0

x0

)dt + f T

i2(x0, xi+1)

uri0

�(t)d�

dzn = dn(t)u pn dt + fn1(x0, x,u)dt + f Tn2(x0, x,u)�(t)d�, 1�i�n−1.

(13)

Remark 3For simplicity, given a vector = (1, . . . ,n), i represents (1, . . . ,i ). In case that no confusionis caused, we sometimes drop the arguments of functions.



To invoke the backstepping method, we define the error variables

ε1 = z1,ε2 = z2 −z∗2(x0, z1,�), . . . ,εn = zn −z∗

n(x0, zn−1,�) (14)

with the virtual smooth control z∗1, . . . , z∗

n defined by

z∗1 =0, z∗

i+1 =−εi�i (x0, zi ,�), i =1,2, . . . ,n−1, (15)

where �i (x0, zi ,�)>0(i =1,2, . . . ,n) are some smooth functions and � is the estimation of �=max{�,�(p0+3)/2,�pi /(pi −2li ),�(p0+3)/(p0−pi +2),�pi (p0+3)/((p0+3−pi )(pi −2li )−pi )|i = 1,2, . . . ,n; li =1,2, . . . , [(pi −1)/3]} with �=‖��T‖∞>0. Then, in the εi -coordinates, the system is changed into

dεk = dk(t)z pkk+1 dt +�k1(x0, zk+1,�)dt +�T

k2(x0, zk+1,�)�d�, 1�k�n−1,

dεn = dn(t)u pn dt +�n1(x0, z,u,�)dt +�Tn2(x0, z,u,�)�d�

(16)

with

�i1(x0, zi+1,�) � fi1

uri0

−ri d0(t)ziu p0

0

x0−d0(t)�p0

0

�z∗i

�x0x p0

0 −i−1∑k=1

�z∗i

�zk

(dk(t)z pk

k+1+fk1

urk0

−rkd0(t)zku p0

0

x0

)−�z∗

i

��˙�−1

2T r

{i−1∑

j,k=1

1

ur j +rk0

�2z∗i

�z j�zk�T f j2 f T

k2�

},

�i2(x0, zi+1,�)� fi2

uri0

−i−1∑k=1

1

urk0

fk2�z∗

i

�zk, i =1, . . . ,n.

Remark 4For convenience, with some abuse of the notations, we denote

∑ij (·)=0 when j>i in the paper.

It can be seen that the nonlinear uncertainties �ij(i =1, . . . ,n; j =1,2) are not available forthe feedback design. Before the systematic controller design procedure is presented, we give thefollowing proposition, which supplies the estimates of these nonlinearities for the coming recursivedesign. The proof of Proposition 1 is given in Appendix A.

Proposition 1For i =1, . . . ,n, there exist nonnegative smooth functions �ij(x0, zi ), �i (x0, zi+1), �ij(x0, zi ), and

�i (x0, zi+1), such that

fi1(x0, xi+1)

uri0

�pi −1∑j=0

|zi+1| j (|ε1|pi − j +·· ·+|εi |pi − j )�ij (17)

�(|ε1|pi +·· ·+|εi |pi +|εi+1|pi )�i , (18)

fi2(x0, xi+1)

uri0

�[(pi −1)/3]∑

j=0|zi+1| j (|ε1|((pi +1)/2)− j +·· ·+|εi |((pi +1)/2)− j )�ij (19)

�(|ε1|(pi +1)/2 +·· ·+|εi |(pi +1)/2 +|εi+1|(pi +1)/2)�i , (20)

fn1(x0, z,u)�pn−1∑j=0

|u| j (|ε1|pn− j +·· ·+|εn|pn− j )�nj , (21)

fn2(x0, z,u)�[(pn−1)/3]∑

j=0|u| j (|ε1|((pn+1)/2)− j +·· ·+|εn|((pn+1)/2)− j )�nj . (22)



After the above analysis, now we turn attention to the controller design.Step 1: Consider the Lyapunov function V1 = (C0/4�)x4

0 +(1/(p0 − p1 +4))ε p0−p1+41 +

(1/2�)�2, where C0 and Ci (i =1,2, . . . ,n) thereafter are positive constants to be determined.

Then, by some direct calculations, we have

LV1 = −C0

��p0

0 d0(t)x p0+30 +d1(t)ε p0−p1+3

1 z p12 +ε

p0−p1+31 �11

+ p0 − p1 +3

2T r{�T�12ε

p0−p1+21 �T

12�}− 1

�� ˙�. (23)

Applying Assumption 3, (14), (15), (17) and Lemma 1, it follows that

εp0−p1+31 �11 �

p1−1∑j=1

j 11p1/j

p1|ε1|p0−p1+3|z2|p1 +ε

p0+31

(�10 +

p1−1∑j=1

p1 − j

p1 11p1/(p1− j)

�p1/(p1− j)1 j

+ p0 − p1 +4

p0 +3(r1��p0

0 |x0|p0−p1 )(p0+3)/(p0−p1+4)

)+ p1 −1

p0 +3x p0+3

0 ,

where 11 and 12, ij(i =2,3, . . . ,n; j =1,2) thereafter are arbitrary positive constants.By (19) and Lemma 1, we have

p0 − p1 +3

2T r{�T�12ε

p0−p1+21 �T

12�}

� p0 − p1 +3

2�ε

p0−p1+21

[(p1−1)/3]∑j,k=0

|ε1|p1+1− j−k |z2|k+ j �T1 j �1k

�[(p1−1)/3]∑

j=1

[(p1−1)/3]∑k=0

2 j

p1

p12 j12 |ε1|p0−p1+3|z2|p1 +ε

p0+31

⎛⎜⎝ [(p1−1)/3]∑k=0

p0 − p1 +3

2��

T10�1k

+[(p1−1)/3]∑

j=1

[(p1−1)/3]∑k=0

p1 −2 j

p1 p1

p1−2 j

12

(p0 − p1 +3

2��

T1 j �1k

) p1p1−2 j

⎞⎟⎠ .

Take the notations

C01 = p1 −1

p0 +3,

�11=[(p1−1)/3]∑

k=0

p0−p1+3

2�

T10�1k+

[(p1−1)/3]∑j=1

[(p1−1)/3]∑k=0

p1−2 j

p1 p1/(p1−2 j)12

(p0−p1+3

2�

T1 j �1k

)p1/(p1−2 j)

,

�12 = �10 +p1−1∑j=1

p1 − j

p1 11

p1p1− j

�p1/(p1− j)1 j + p0 − p1 +4

p0 +3(r1��p0

0 x p0−p10 )(p0+3)/(p0−p1+4),

where C01�0 is a constant, �11 and �12 are nonnegative smooth functions. Choose 11, 12

satisfying∑p1−1

j=1j 11

p1/j

p1+∑[(p1−1)/3]

j=1

∑[(p1−1)/3]k=0

2 jp1

p1/2 j12 ��/2, then we obtain

LV1 � −(C0�p00 −C01)x p0+3

0 +d1(t)ε p0−p1+31 z p1

2 + �

2|ε1|p0−p1+3|z2|p1

+εp0+31 (��11 +�12)− 1

�� ˙�.



In order to deal with the terms d1(t)ε p0−p1+31 z p1

2 + �2 |ε1|p0−p1+3|z2|p1 , we have the following

proposition, whose proof can be found in Appendix B.

Proposition 2For i =1,2, . . . ,n, the following statement holds true:

di (t)εp0−pi +3i z pi

i+1 + �

2|εi |p0−pi +3|zi+1|pi �

(�+ �

2

)|εi |p0−pi +3|z pi

i+1 −z∗pii+1|+

�

2ε

p0−pi +3i z∗pi

i+1.

Then, by a direct application of Proposition 2, we have

LV1 � −(C0�p00 −C01)x p0+3

0 +εp0−p1+31

(�

2z∗p1

2 +εp11 (�11

√1+�

2 +�12)

)

+(

�+ �

2

)|ε1|p0−p1+3|z p1

2 −z∗p12 |+ 1

��(�ε p0+3

1 �11 − ˙�).

Take 1 =�ε p0+31 �11 and select the virtual control z∗

2 as

z∗2 =−ε1

⎛⎝2C1 +�11

√1+�

2 +�12

�

⎞⎠1/p1

�−ε1�1(x0, z1,�), (24)

and this leads to

LV1�−(C0�p00 −C01)x p0+3

0 −C1εp0+31 + 1

��(1 − ˙�)+

(�+ �

2

)|ε1|p0−p1+3|z p1

2 −z∗p12 |. (25)

Step i (2�i�n): Suppose at Step i-1, there exists a positive definite and proper Lyapunov function

Vi−1 = (C0/4�)x40 +∑i−1

k=1 (1/(p0 − pk +4))ε p0−pk+4k +(1/2�)�

2, which satisfies

LVi−1 � −(

C0�p00 −

i−1∑k=1

C0k

)x p0+3

0 −i−1∑k=1

(Ck −

i−1∑j=k+1

� j

)ε

p0+3k +(i−1 − ˙�)

·(

1

��+

i−1∑k=2

εp0−pk+3k

�z∗k

��

)+(

�+ �

2

)|εi−1|p0−pi−1+3|z pi−1

i −z∗pi−1i |. (26)

In the following, it will be shown that a similar property to (26) also holds at Step i.Choose the Lyapunov function Vi =Vi−1 +(1/(p0 − pi +4))ε p0−pi +4

i , and we have

LVi =LVi−1 +di (t)εp0−pi +3i z pi

i+1 +εp0−pi +3i �i1 + p0 − pi +3

2T r{�T�i2ε

p0−pi +2i �T

i2�}

� −(

C0�p00 −

i−1∑k=1

C0k

)x p0+3

0 −i−1∑k=1

(Ck −

i−1∑j=k+1

� j

)ε

p0+3k +di (t)ε

p0−pi +3i z pi

i+1

+(

�+ �

2

)|εi−1|p0−pi−1+3|z pi−1

i −z∗pi−1i |+ p0 − pi +3

2T r{�T�i2ε

p0−pi +2i �T

i2�}

+εp0−pi +3i �i1 +(i−1 − ˙�)

(1

��+

i−1∑k=2

εp0−pk+3k

�z∗k

��

). (27)



The following proposition is provided to estimate the uncertain terms in (27). The proof is trivialbut complicated, and can be found in Appendix C.

Proposition 3Let i =i−1 +�ε p0+3

i �i1, then there exist nonnegative constants C0i , �i and nonnegative functions�i1, �i2 such that(

�+ �

2

)|εi−1|p0−pi−1+3|z pi−1

i −z∗pi−1i |+ p0 − pi +3

2T r{�T�i2ε

p0−pi +2i �T

i2�}

+εp0−pi +3i �i1 +(i−1 − ˙�)

(1

��+

i−1∑k=2

εp0−pk+3k

�z∗k

��

)

�C0i x p0+30 +�i

i−1∑k=1

εp0+3k + �

2|εi |p0−pi +3|zi+1|pi +ε

p0+3i (�i1�+�i2)

+(i − ˙�)

(1

��+

i∑k=2

εp0−pk+3k

�z∗k

��

).

From Propositions 2 and 3, it can be concluded that

LVi � −(

C0�p00 −

i∑k=1

C0k

)x p0+3

0 −i−1∑k=1

(Ck −

i∑j=k+1

� j

)ε

p0+3k +(i − ˙�)

·(

1

��+

i∑k=2

εp0−pk+3k

�z∗k

��

)+(

�+ �

2


i+1 −z∗pii+1|

+εp0−pi +3i

(�

2z∗pi

i+1 +εpii (�i1

√1+�

2 +�i2)

).

Select z∗i+1 as

z∗i+1 =−εi

⎛⎝2Ci +�i1

√1+�

2 +�i2

�

⎞⎠1/pi

�−εi�i (x0, z1, . . . , zi ,�) (28)

and a direct substitution results in

LVi � −(

C0�p00 −

i∑k=1

C0k

)x p0+3

0 −i∑

k=1

(Ck −

i∑j=k+1

� j

)ε

p0+3k +(i − ˙�)

·(

1

��+

i∑k=2

εp0−pk+3k

�z∗k

��

)+(

�+ �

2


i+1 −z∗pii+1|. (29)

Particularly, when i =n, we get the controller u = zn+1 = z∗n+1 and the parameter update law:

u =−εn

⎛⎝2Cn +�n1

√1+�

2 +�n2

�

⎞⎠1/pn

�−εn�n(x0, z1, . . . , zn,�), (30)

˙�=n =n−1 +�ε p0+3n �n1, (31)



such that the Lyapunov function

Vn = C0

4�x4

0 +n∑

k=1

1

p0 − pk +4ε

p0−pk+4k + 1

2��

2

satisfies

LVn�−(

C0�p00 −

n∑k=1

C0k

)x p0+3

0 −n∑

k=1

(Ck −

n∑j=k+1

� j

)ε

p0+3k . (32)

4. SWITCHING CONTROL STRATEGY

In Section 3, suppose x0(t0) �=0, we have designed the controllers (9) and (30) for system (1). Nowwe turn to the case of x0(t0)=0. Without loss of generality, it is assumed that t0 =0.

Considering the starting point of the x0-subsystem is zero, we can first choose a non-zeroconstant action u0 =u∗

0 to drive the state x0 away from zero. Then, at a time instant t∗s >0, thestate x0(t∗s ) �=0, and the developed design scheme in Section 3 can be applied. Along this line, wefirst choose the following constant control:

u0 =u∗0, u∗

0>0. (33)

Substituting (33) into the x0-subsystem of (1), we can obtain

x0 =d0(t)u p00 >�u∗p0

0 >0, x0(0)=0. (34)

From (34), it is known that for any small enough constant c∗0, there exists an instant t∗s such that

x0(t∗s )=c∗0. Since u0 =u∗

0>0 is a constant in the finite time interval [0, t∗s ), the transformation (11) isnot needed before the backstepping design. Then similar to Section 3, we can design the controlleru =u∗(x0, x,�) in [0, t∗s ), which guarantees the state of system (1) is bounded in probability duringthe time period [0, t∗s ). At the time instant t∗s >0, considering the state x0(t∗s )=c∗

0>0, we can switchthe control inputs u0 and u into (9) and (30), respectively.

Based on the aforementioned analysis, now we state the main results of this paper.

Theorem 1Suppose that Assumptions 1–3 hold. If the proposed control design procedure together with theabove switching control strategy is applied to system (1), then, for any initial conditions in thestate space �={(x0, x)∈Rn+1}, system (1) will be almost asymptotically stabilized in proba-bility at the equilibrium and specifically, the state is globally asymptotically regulated to zero inprobability.

ProofLet us first consider the case that the initial state belongs to �0 ={(x0, x)∈Rn+1|x0 �=0}⊂�.As mentioned already, under the control law (9), x0 is asymptotically convergent to zero butwould not be zero for t�0, which guarantees that the transformation (11) is well defined. In (32),choose C0 and Ck such that C0�

p00 −∑n

k=1 C0k>0, Ck −∑nj=k+1 � j>0(k =1,2, . . . ,n), then from

Lemma 2, we conclude that the signals �, ε1, . . . ,εn are bounded in probability and ε1, . . . ,εnconverge to zero in probability. Consequently, z1 =ε1 is bounded in probability and convergesto zero in probability. Since z∗

2(x0, z1,�) is a smooth function of x0, z1,� and �=�−� isbounded in probability, it is known that z∗

2 and z2 =ε2 +z∗2 =ε2 −ε1�1 are bounded in probability



and converge to zero in probability. Similarly, by a recursive argument, these properties also holdfor z3, . . . , zn . As a result of (11), x0, x1, . . . , xn are bounded in probability and converge to zeroin probability. This shows that the feedback laws in the original coordinate are both well definedand bounded in probability in �0. Therefore, the system is almost asymptotically stabilized inprobability.

When x0(0)=0, we use the constant control (33) rendering x0 far away from the origin. Mean-while, by application of the design procedure proposed in Section 3, we construct a controlleru =u∗(x0, x,�), which guarantees that all the signals are bounded in probability during [0, t∗s ).Then, in view of x0(t∗s ) �=0, apply the switching control strategy to system (1) at the time instantt∗s >0. This completes the proof. �

If the state xi+1 and input u do not appear in the drift nonlinearities fij(i =1, . . . ,n−1; j =1,2)and fn j ( j =1,2), the system (1) will degenerate into the following form (35), which is a subclassof system (1). A straightforward application of the proposed methodology in the paper can yieldthe same results with Theorem 1.

Corollary 1For the following class of high order stochastic nonholonomic system:


dxi = di (t)xpii+1uqi

0 dt + fi1(x0, xi )dt + f Ti2(x0, xi )�(t)d�, 1�i�n−1,

dxn = dn(t)u pn dt + fn,1(x0, x)dt + f Tn,2(x0, x)�(t)d�,

(35)

where p0, p1, . . . , pn are odd integers, q1,q2, . . . ,qn−1 are positive integers. If Assumptions 1–3hold, then there exist state-feedback laws to render the system almost asymptotically stabilized atthe origin and the state globally asymptotically regulated to zero in probability. Here, Assumption 1is simplified as that, there exist smooth nonnegative functions �i , �i satisfying

fi1(x0, xi )�(|x1|pi +|x2|pi +·· ·+|xi |pi )�i (x0, xi ),

fi2(x0, xi )�(|x1|pi +1

2 +|x2|pi +1

2 +·· ·+|xi |pi +1

2 )�i (x0, xi ), i =1, . . . ,n.

In [14], under some conditions, the almost asymptotic stabilization can be achieved for thefollowing multi-input power chained form system:

x0 = u p00

x11 = x p1112 uq11

0 · · · xm1 = x pm1m2 uqm1

0

...

x1,n1−1 = xp1,n1−1

1,n1u

q1,n1−1

0 · · · xm,nm−1 = xpm,nm−1m,nm u

qm,nm−10

x1,n1 =u p1,n1 · · · xm,nm =u pm,nm ,

(36)

where x = (x0, XT1 , . . . , XT

m)T ∈RN with N =1+∑mi=1 ni , and Xi = (xi1, . . . , xi,ni )

T for 1�i�m,u = (u0,u1, . . . ,um)T ∈Rm+1 represent the state and the control input, respectively, p0, pl,lm areodd integers, ql,lv are positive integers, l =1,2, . . . ,m; lm =1,2, . . . ,nl ; lv =1,2, . . . ,nl −1.

It should be noted that the obtained results in this paper can be generalized to a wider class ofnonholonomic systems than (36). In fact, we have the following corollary.



Corollary 2For a class of multi-input high order nonholonomic systems of the form


dx11 = d11(t)x p1112 uq11

0 dt + f1,11(x0, x11, x12)dt + f T1,12(x0, x11, x12)�(t)d�,

...

dx1,n1−1 = d1,n1−1(t)xp1,n1−1

1,n1u

qn1−1

0 dt + f1,n1−1,1(x0, x1,n1 )dt + f T1,n1−1,2(x0, x1,n1 )�(t)d�,

dx1,n1 = d1,n1 (t)u1p1,n1 dt + f1,n1,1(x0, x1,n1,u1)dt + f T

1,n1,2(x0, x1,n1,u1)�(t)d�,

...

dxk1 = dk1(t)x pk1k2 uqk1

0 dt + fk,11(xk0, xk1, xk2)dt + f Tk,12(xk0, xk1, xk2)�(t)d�,

...

dxk,nk−1 = dk,nk−1(t)xpk,nk−1

k,nku

qnk−1

0 dt + fk,nk−1,1(x0, xk,nk )dt + f Tk,nk−1,2(x0, xk,nk )�(t)d�,

dxk,nk = dk,nk (t)ukpk,nk dt + fk,nk ,1(x0, xk,nk ,uk)dt + f T

k,nk ,2(x0, xk,nk ,uk)�(t)d�,

(37)

where (x0, xT)T = (x0, x11, . . . , x1,n1, . . . , xi1, . . . , xini , . . . , xk1, . . . , xk,nk )T ∈RN with N =1+∑ki=1 ni is the system state and u = (u0,u1, . . . ,uk)T are inputs, p0 and pl,lm are odd integers,

ql,lv are positive integers (l =1,2, . . . ,k; lm =1,2, . . . ,nl; lv =1,2, . . . ,nl −1). For every subsystem(x0, xi1, . . . , xi,ni )(i =1,2, . . . ,k), if Assumptions 1–3 are satisfied, then there exist control lawsu0,u1,u2, . . . ,uk such that the whole system is stabilized at the origin in probability and the statesare globally asymptotically regulated to zero in probability.

5. SIMULATION EXAMPLE

In this section, a simulation example is provided to illustrate the effectiveness of the developeddesign scheme. Consider the following nonholonomic systems:

dx0 = u30 dt,

dx1 = x32u2

0 dt + 14 x1x2

2 dt + 14 x2

1�d�,

dx2 = u3 dt + 18 x2

2�d�,

(38)

where (x0, xT)T = (x0, x1, x2)T is the system state, u0 and u are the control inputs, � is an one-dimensional independent standard Winer process with incremental covariance �(t)�(t)T dt , i.e.E{d�d�T}=�(t)�(t)T dt , where �(t) is a bounded nonnegative function for each t . It can beseen that system (38) is in the form of (1) with p0 = p1 = p2 =3, q1 =2, f11(x0, x1, x2)= 1

4 x1x22 ,

f12(x0, x1, x2)= 14 x2

1 , f21(x0, x1, x2,u)=0, and f22(x0, x1, x2,u)= 18 x2

2 . Denote ‖��T‖∞>0 as �

and � as the estimation of �=max{�,�3}.We start from the case of x0(0)=0. First, choose u∗

0 =1 and c∗0 =1. Then there exists t∗s >0

satisfying x0(t∗s )=1. Using the proposed algorithm in Section 3, we have

u =−ε2(0.5+�21

√1+�

2 +�22)1/3, (39)

˙�=2 (40)



0 10 20 30

0

0.5

1

1.5x

0

0 10 20

0

x1

0 10 20 30

0

0.2

0.4x

2

0 5 100.9999

1

1.0001

0 10 20

0

0.5

1

1.5u

0

0 5 10

0

10

20

30u

Figure 1. Responses of the closed-loop system.

with �21= 38+ 25

8 �121 + 25

64 (�x∗2/�x1)6, �22=50.625+6335.3�12

1 +80(�x∗2/�x1)2�6

1+4√

1+(�x∗2/�x1)2

+ε22�21

√1+ε2

2(�x∗2/��)2 + 45

1024ε61(�x∗

2/��)2, 2 = 332ε6

1 +ε62�21, ε1 = x1, ε2 = x2 −x∗

2 , x∗2 =

−ε1�1, �1 = (2(2+ 332

√1+�

2))1/3, �x∗

2/�x1 =−�1, �x∗2/��=− 1

16ε1�−21 (�/

√1+�

2).

Then due to x0(t∗s )=1 �=0, we choose the control law u0 =−2x0 in the first subsystem of (38).Again, by using the developed design procedure in Section 3, we can obtain that

u =−ε2(0.5+�21

√1+�

2 +�22)13 , (41)

˙�=2 (42)

with �21 = 38 + 25

8 �121 +1600(�z∗

2/�z1)6, �22 =50.625+6335.3�121 +8x2

0 (�z∗2/�x0)2 +80(�z∗

2/�z1)2

�61 +0.54(�z∗

2/�z1)65 +4

√1+(�z∗

2/�z1)2 +16ε21(�z∗

2/�z1)2 +2.5�41(�z∗

2/�z1)2 +5ε21x8

0 (�2z∗2/�z2

1)2

+5ε61�

211(�z∗

2/��)2 +ε22�21

√1+ε2

2(�z∗2/��)2, 2 =ε6

1�11 +ε62�21, z1 = x1/(−2x0)3, z2 = x2, ε1 =

z1, ε2 = z2 −z∗2, z∗

2 =−ε1�1, �1 = (2(2+8ε21 +�11

√1+�

2))1/3, �11 = 3

2 x40 , �z∗

2/�x0 =−4ε1x30�−2

1√1+�

2, �z∗

2/�z1=−�1− 323 ε2

1�−21 , �z∗

2/��=− 23ε1�

−21 �11(�/

√1+�

2), �2z∗

2/�z21 =−32ε1�

−21 +

329 ×64ε3

1�−51 . The simulation result shown in Figure 1 is carried out with the initial condition

(x0(0), x1(0), x2(0))= (0,−0.2,0.1).

6. CONCLUSION

In this paper, we investigate the global stabilization problem for a class of high order nonholonomicsystems. With the help of backstepping technique, a systematic control design procedure is devel-oped in the stochastic setting. To get around the stabilization burden associated with nonholonomicsystems, a switching control strategy is proposed. It is shown that the designed control laws canguarantee that the closed-loop system is almost asymptotically stabilized in probability and thestate is globally asymptotically regulated to zero in probability. In addition, the proposed approachcan be applied to the multi-input high order nonholonomic system with the stochastic disturbances.The obtained result is a further extension of the existing works.



APPENDIX A: PROOF OF PROPOSITION 1

For sparing the space, only the proof of (17) (18) is provided here, and the others can be provedin the same manner. Indeed, from (9) and Assumption 1, we get

fi1(x0, xi+1)

uri0

�pi −1∑j=0

i∑k=1

|zi+1| j |zk |pi − j (�0|x0|)rk (pi − j)+ jri+1−ri �ij(x0, zi ).

In terms of (12), it is evident that

rk(pi − j)+ jri+1−ri�0, k =1,2, . . . , i; j =0,1, . . . , pi −1.

Considering x0 is bounded, we can find a suitable nonnegative smooth functions �ij such that

fi1(x0, xi+1)

uri0

�pi −1∑j=0

|zi+1| j (|z1|pi − j +|z2|pi − j +·· ·+|zi |pi − j )�ij(x0, zi ).

From (14) and (15), it can be verified that

fi1(x0, xi+1)

uri0

�pi −1∑j=0

i∑k=1

2pi − j−1|zi+1| j (|εk |pi − j +|εk−1�k−1|pi − j )�ij(x0, zi )

�pi −1∑j=0

i∑k=1

|zi+1| j |εk |pi − j 2pi − j−1(1+�kpi − j )�ij(x0, zi ).

Denote �ij�2pi − j−1(1+∑ik=1 �k

pi − j )�ij(x0, zi ), then we have (17). Furthermore,

fi1(x0, xi+1)

uri0

�pi −1∑j=0

i∑k=1

2 j−1(|εi+1| j +|εi�i | j )|εk |pi − j �ij

�pi −1∑j=0

i∑k=1

2 j−1(

j

pi|εi+1|pi + j

pi|εi�i |pi + 2(pi − j)

pi|εk |pi

)�ij.

Also, one can find nonnegative smooth functions �i (·) satisfying (18). This completes the proof.

APPENDIX B: PROOF OF PROPOSITION 2

For i=1,2, . . .,n, with the choice of �i (·)>0 and z∗i+1=−εi�i (x0, zi ,�), we have −ε

p0−pi +3i

z∗pii+1�0. Therefore, from Assumption 3, it can be deduced in the following way:

di (t)εp0−pi +3i z pi

i+1 + �

2|εi |p0−pi +3|zi+1|pi

=(

di (t)+ �

2sgn(ε p0−pi +3

i z pii+1)

)ε

p0−pi +3i z pi

i+1 − �

2ε

p0−pi +3i z∗pi

i+1 + �

2ε

p0−pi +3i z∗pi

i+1

�(

di (t)+ �

2sgn(ε p0−pi +3

i z pii+1)

)ε

p0−pi +3i (z pi

i+1 −z∗pii+1)+ �

2ε

p0−pi +3i z∗pi

i+1

�(

�+ �

2


i+1 −z∗pii+1|+

�

2ε

p0−pi +3i z∗pi

i+1.

Then the proof of Proposition 2 is completed.



APPENDIX C: PROOF OF PROPOSITION 3

Noting that |x p − y p|�p |x − y||x p−1 + y p−1|, x, y ∈ R with any positive odd integer p. From (14),(15) and Lemma 1, it is not difficult to verify that(

�+ �

2

)|εi−1|p0−pi−1+3|z pi−1

i −z∗pi−1i |

�2p0 − pi−1 +5

p0 +3ε

p0+3i−1 +ε

p0+3i

pi−1

p0 +3

(2pi−1−2 pi−1

(�+ �

2

))(p0+3)/pi−1

+εp0+3i

1

p0 +3

(pi−1(2pi−1−2 +1)

(�+ �

2

)�pi−1−1

i−1

)p0+3

.

By (17) and Lemma 1, we can obtain

εp0−pi +3i

fi1

uri0

�pi −1∑j=1

i j i1pi /j

pi|εi |p0−pi +3|zi+1|pi + p2

i

p0 +3

i−1∑k=1

εp0+3k

+εp0+3i

⎛⎝ p2i

p0 +3+

i∑k=1

p0 − pi +3

p0 +3�(p0+3)/(p0−pi +3)

i0

+pi −1∑j=1

i∑k=1

p0 − pi +3

p0 +3

(pi − j

pi i1

pipi − j

�pi

pi − j

ij

)(p0+3)/(p0−pi +3)⎞⎠ .

According to Assumption 3 and Lemma 1, from (14) (15), we have

ri�p00 d0(t)ziε

p0−pi +3i x p0−1

0

�2(p0 −1)

p0 +3x p0+3

0 + 1

4ε

p0+3i−1 +ε

p0+3i

⎛⎝4(ri��p00 ε

p0−pii )(p0+3)/4

p0 +3

+3

4

(4(ri��p0

0 �i−1)(p0+3)/4

p0 +3

)4/3

|εi |(p0+3)(p0−pi )

3

⎞⎠ ,

−i−1∑k=1

dk(t)�z∗

i

�zkε

p0−pi +3i z pk

k+1

�εp0+3i

i−1∑k=1

p0 +3− pk

p0 +3

(2pk−1�ε

pk−pii �pk

k

∣∣∣∣�z∗i

�zk

∣∣∣∣)(p0+3)/(p0+3−pk )

+εp0+3i 2pi−1−1�ε

pi−1−pii

∣∣∣∣ �z∗i

�zi−1

∣∣∣∣+ i−1∑k=1

pk + pk−1

p0 +3ε

p0+3k

+εp0+3i

i−1∑k=2

p0 +3− pk−1

p0 +3

(2pk−1−1�ε

pk−1−pii

∣∣∣∣ �z∗i

�zk−1

∣∣∣∣)(p0+3)/(p0+3−pk−1)

,

−i−1∑k=1

rk�p00 d0(t)

�z∗i

�zkzkε

p0−pi +3i x p0−1

0



�2(i −1)(p0 −1)

p0 +3x p0+3

0 + 1

2

i−1∑k=1

εp0+3k +ε

p0+3i

i−1∑k=1

⎛⎜⎜⎜⎝3×41/3

(rk��p0

0 εp0−pii

∣∣∣∣�z∗i

�zk

∣∣∣∣)(p0+3)/3

(p0 +3)4/3

+3×41/3

(rk��p0

0 εp0−pii �k−1

∣∣∣∣�z∗i

�zk

∣∣∣∣)(p0+3)/3

(p0 +3)4/3

⎞⎟⎟⎟⎠ .

By using Lemma 1, there holds

−d0(t)�p00

�z∗i

�x0x p0

0 εp0−pi +3i � p0

p0 +3x p0+3

0 +εp0+3i

3

p0 +3

(��p0

0 εp0−pii

∣∣∣∣�z∗i

�x0

∣∣∣∣)(p0+3)/3

.

In view of (18) and Lemma 1, it can be seen that

−i−1∑k=1

1

urk0

εp0−pi +3i fk1

�z∗i

�zk

�i−1∑k=1

(i−1∑

l=k−1

pl

p0 +3

)ε

p0+3k

+εp0+3i

(pi−1

p0 +3+

i−1∑k=1

k+1∑l=1

p0 +3− pk

p0 +3

(ε

pk−pii �k

∣∣∣∣�z∗i

�zk

∣∣∣∣)(p0+3)/(p0+3−pk )).

Again, using Lemma 1, in terms of (19) (20), we have

−1

2ε

p0−pi +3i T r

{i−1∑

j,k=1

1

ur j +rk0

�2z∗i

�z j�zk�T f j2 f T

k2�

}

�i−1∑k=1

(i−1∑

l=k−1

pl +1

p0 +3

)ε

p0+3k +ε

p0+3i

(i−1∑j=1

j +1

2�|εi |pi−1−pi +1�

Ti−1� j

∣∣∣∣∣ �2z∗i

�z j�zi−1

∣∣∣∣∣+

i−1∑k=1

i−1∑l=k−1

p0 +2− pl

p0 +3

(i−1∑j=1

j +1

2�|εi |pl−pi +1�

Tl � j

∣∣∣∣∣ �2z∗i

�z j�zl

∣∣∣∣∣)(p0+3)/(p0+2−pl )

⎞⎠and

p0 − pi +3

2ε

p0−pi +2i T r{�T�i2�

Ti2�}

�[(pi −1)/3]∑

j=1

[(pi −1)/3]∑k=0

i∑l=1

2 j

pi pi /2 j

i2 |εi |p0−pi +3|zi+1|pi

+i−1∑k=1

([(pi −1)/3]∑

j,k=0

pi + pipi −2 j

p0 +3+ i2(pi +1)

p0 +3

)ε

p0+3k

+εp0+3i

((p0 − pi +3)�

i−1∑j=1

( j +1)ε pi−1−pii

∣∣∣∣�z∗i

�z j

∣∣∣∣ ∣∣∣∣�z∗i

�zk

∣∣∣∣ �Ti−1� j



+i−1∑k=1

i−1∑l=k−1

i−1∑j=1

p0 +2− pi

(p0 +3)

(( j +1)(p0 − pi +3)�ε pl−pi

k �Tl � j

∣∣∣∣�z∗i

�z j

∣∣∣∣ ∣∣∣∣�z∗i

�zk

∣∣∣∣)(p0+3)/(p0+2−pi )

+[(pi −1)/3]∑

j=1

[(pi −1)/3]∑k=0

pi −2 j

pi pi /(pi −2 j)i2

(i(p0 − pi +3)��Tij �ik)pi /(pi −2 j)

+[(pi −1)/3]∑

k=0i(p0 − pi +3)��

Ti0�ik

+[(pi −1)/3]∑

k=0

i−1∑l=1

p0 +2− pi

p0 +3(i(p0 − pi +3)��

Ti0�ik)(p0+3)/(p0+2−pi )

+[(pi −1)/3]∑

j=1

[(pi −1)/3]∑k=0

i−1∑l=1

p0 +3− pi − pi

pi −2 jp0 +3

×(

pi −2 j


(i(p0 − pi +3)��Tij �ik)pi /(pi −2 j)

)(p0+3)/(p0+3−pi −(pi /(pi −2 j)))⎞⎠ .

Choose positive constants i1 and i2 such that∑pi −1

j=1i j i1

pi /j

pi+∑[(pi −1)/3]

j=1

∑[(pi −1)/3]k=0∑i

l=12 jpi

pi /2 ji2 ��

2 .Define the nonnegative smooth function �i1(·) as follows:

�i1 =i−1∑k=1

i−1∑l=k−1

p0+2−pl

p0+3

⎛⎜⎝i−1∑j=1

j+1

2�

Tl � j

√√√√1+(

εpl−pi +1i

�2z∗i

�z j�zl

)2⎞⎟⎠

(p0+3)/(p0+2−pl )

+[(pi −1)/3]∑

k=0

i−1∑l=1

p0+2−pi

p0+3(i(p0−pi+3)�

Ti0�ik)(p0+3)/(p0+2−pi )+

[(pi −1)/3]∑j=1

[(pi −1)/3]∑k=0

i−1∑l=1

p0+3−pi− pi

pi−2 jp0+3

(pi−2 j


(i(p0−pi+3)�Tij �ik)pi /(pi −2 j)

)(p0+3)/(p0+3−pi −(pi /(pi −2 j)))

+i−1∑k=1

i−1∑l=k−1

i−1∑j=1

p0+2−pi

p0+3

⎛⎜⎝( j+1)(p0−pi+3)ε pl−pik

√√√√1+(

�z∗i

�z j

�z∗i

�zk

)2

· �Tl � j

)(p0+3)/(p0+2−pi )

+i−1∑j=1

(p0−pi+3)( j+1)ε pi−1−pii �

Ti−1� j

√√√√1+(

�z∗i

�z j

�z∗i

�zk

)2

+i−1∑j=1

j+1

2�

Ti−1� j

√√√√1+(

εpi−1−pi +1i

�2z∗i

�z j�zi−1

)2

+[(pi −1)/3]∑

k=0i(p0−pi+3)��

Ti0�ik

+[(pi −1)/3]∑

j=1

[(pi −1)/3]∑k=0

pi−2 j


(i(p0−pi+3)��Tij �ik)pi /(pi −2 j).



In view of i =i−1 +�ε p0+3i �i1, by Lemma 1, after some direct calculations, we can obtain

εp0−pi +3i

�z∗i

��i �

i−1∑k=1

pi

p0+3ε

p0+3k +ε

p0+3i

(i−1∑k=1

p0+3−pi

p0+3

(�|εk |p0−pi +3�k1

∣∣∣∣�z∗i

��

∣∣∣∣)(p0+3)/(p0+3−pi )

+�|εi |p0−pi +3�i1

∣∣∣∣�z∗i

��

∣∣∣∣)

and

εp0−pi +3i

�z∗i

��(i − ˙�)+ε

p0+3i �i1�+(i−1 − ˙�)

(1

��+

i−1∑k=2

εp0−pk+3k

�z∗k

��

)

�(i − ˙�)

(1

��+

i∑k=2

εp0−pk+3k

�z∗k

��

)+ε

p0+3i ��i1

√√√√1+(

i−1∑k=2

εp0−pk+3k

�z∗k

��

)2

.

Introduce the following notations:

C0i = 2(i+1)p0−(2i+1)

p0+3,

�i = 3

4+5+2p0+2p1+p2

i +i2(pi+1)

p0+3+

i−1∑l=1

2pl+1

p0+3+

[(pi −1)/3]∑j,k=0

pi+ pipi −2 j

p0+3,

�i2 =pi−1

(2pi−1−2 pi−1

(�+�

2

))(p0+3)/(pi−1)

p0+3+

(pi−1(2pi−1−2+1)

(�+�

2

)�pi−1−1

i−1

)p0+3

p0+3

+i∑

k=1

p0−pi+3

p0+3�(p0+3)/(p0−pi +3)

i0 +pi −1∑j=1

i∑k=1

p0−pi+3

p0+3

×(

pi− j

pi i1pi /(pi − j)

�pi /(pi − j)ij

)(p0+3)/(p0−pi +3)

+3

⎛⎝��p00 ε

p0−pii

√1+

(�z∗

i

�x0

)2⎞⎠(p0+3)/3

p0+3

+6i−1∑k=1

⎛⎝rk��p00 ε

p0−pii

√1+

(�z∗

i

�zk

)2⎞⎠(p0+3)/3

(p0+3)4/3

+6i−1∑k=1

⎛⎝rk��p00 ε

p0−pii �k−1

√1+

(�z∗

i

�zk

)2⎞⎠(p0+3)/3

(p0+3)4/3+��i1

√√√√1+(

i−1∑k=2

εp0−pk+3k

�z∗k

��

)2



+i−1∑k=1

p0+3−pi

p0+3

⎛⎝��k1

√1+

(ε

p0−pi +3k

�z∗i

��

)2⎞⎠(p0+3)/(p0+3−pi )

+��i1

√1+

(ε

p0−pi +3i

�z∗i

��

)2

+i−1∑k=1

k+1∑l=1

p0+3−pk

p0+3

⎛⎝εpk−pii �k

√1+

(�z∗

i

�zk

)2⎞⎠(p0+3)/(p0+3−pk )

+4(ri��p00 ε

p0−pii )(p0+3)/4

p0+3

+i−1∑k=2

p0+3−pk−1

p0+3

⎛⎝2pk−1−1�εpk−1−pii

√1+

(�z∗

i

�zk−1

)2⎞⎠(p0+3)/(p0+3−pk−1)

+ pi−1+p2i

p0+3

+2pi−1−1�εpi−1−pii

√1+

(�z∗

i

�zi−1

)2

+11(ri��p00 �i−1ε

p0−pii )(p0+3)/3

(p0+3)4/3

+i−1∑k=1

p0+3−pk

p0+3

⎛⎝2pk−1�εpk−pii �pk

k

√1+

(�z∗

i

�zk

)2⎞⎠(p0+3)/(p0+3−pk )

,

where C0i , �i are optional nonnegative constants and �i2 is a nonnegative smooth function. Then,by some simple combinations of the above estimates, Proposition 3 can be hence established.

ACKNOWLEDGEMENTS

The work received support from National Natural Science Foundation of China (grant numbers: 60674027,60974127, 60904022) and The Key Project of Education Ministry of China (grant number: 208074).

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state-feedback stabilization for a class of more general high order stochastic nonholonomic systems

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