state-feedback stabilization for stochastic high-order nonlinear systems with siss inverse dynamics

Asian Journal of Control, Vol. 14, No. 1, pp. 207 216, January 2012Published online 18 October 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.288

STATE-FEEDBACK STABILIZATION FOR STOCHASTIC HIGH-ORDER

NONLINEAR SYSTEMS WITH SISS INVERSE DYNAMICS

Liang Liu and Xue-Jun Xie

ABSTRACT

This paper investigates the problem of state-feedback control for a classof stochastic high-order nonlinear systems with stochastic inverse dynamics.Under the assumption that the inverse dynamics of the subsystem are stochasticinput-to-state stable (SISS), by extending through adding a power integratortechnique, choosing an appropriate Lyapunov function and using the ideaof changing supply function, a smooth state-feedback controller is explicitlyconstructed to render the system globally asymptotically stable in probabilityand the states can be regulated to the origin. A simulation example is providedto show the effectiveness of the proposed scheme.

Key Words: Stochastic high-order nonlinear systems, stochastic inversedynamics, stochastic input-to-state stability (SISS), state-feedback control.

I. INTRODUCTION

Consider the following stochastic high-ordernonlinear systems with stochastic inverse dynamics

dz = f0(z, x1)dt+g0(z, x1)d�,

dx1 = (x p2 + f1(z, x1))dt+g1(z, x1)d�,

...

dxn−1 = (x pn + fn−1(z, xn−1))dt

+gn−1(z, xn−1)d�,

dxn = (u p+ fn(z, x))dt+gn(z, x)d�, (1)

Manuscript received January 27, 2010; revised June 6, 2010;accepted July 30, 2010.The authors are with the Institute of Automation, Qufu

Normal University, Shandong Province, 273165, China (e-mail:[email protected], [email protected]).This work is supported by National Natural Science Foun-

dation of China (No.60774010, 10971256), Shandong Provin-cial Natural Science Foundation of China (No.ZR2009GM008,ZR2009AL014).

where z∈Rm is the unmeasurable stochastic inversedynamics, x=(x1, . . . , xn)T ∈Rn and u∈R are themeasurable system state and the control input, respec-tively. xi =(x1, . . . , xi )T , i=1, . . . ,n, xn = x , p∈R∗�{q∈R :q= n

m ≥1 for any positive odd integersm,n} is said the high-order of system, � is an r -dimensional standard Wiener process defined on acomplete probability space (�,F, P) with � beinga sample space, F being a filtration, and P being aprobability measure. f0 :Rm×R→Rm , g0 :Rm×R→Rm×r , fi :Rm×Ri →R and gi :Rm×Ri →R1×r areassumed to be smooth functions with fi (0,0)=0 andgi (0,0)=0, i=1, . . . ,n.

In recent years, the study of various problemson stochastic nonlinear systems has achieved remark-able development, see [1–7] and references therein.When p=1, the feedback controller design problemof stochastic nonlinear systems with stochastic inversedynamics has been extensively developed, e.g. [1–5]and references therein. However, all these mentionedcontrollers are only robust against the inverse dynamicswith stringent stability margin. To weaken the stringentcondition on the inverse dynamics, [8] firstly introducedthe concept of stochastic input-to-state stability (SISS).Then, [9] gave a sufficient condition using Lyapunov

q 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

208 Asian Journal of Control, Vol. 14, No. 1, pp. 207 216, January 2012

function on SISS and investigated the problem ofdecentralized adaptive output-feedback stabilizationcontrol for large-scale stochastic nonlinear systems,[10] studied stochastic input-to-state stable and itsapplication to stability of cascaded stochastic nonlinearsystems.

Motivated by the powerful tool for fruitful deter-ministic results in [11, 12] and the related papers, Xieand his co-authors addressed different control problemsfor systems (1) with different structures. When z=0,[13–16] considered respectively the state-feedbackstabilization problem for more general systems withdifferent structures, [17, 18] considered the output-feedback stabilization and the inverse optimal stabi-lization for stochastic high-order nonlinear systems,respectively. In recent results, [19] achieved the output-feedback stabilization under the weaker assumptionson the high-order and the drift and diffusion terms,[20] removed the restrictions on the high-order and thedrift and diffusion terms at the expenses of the outputtracking. For the case of z �=0, only one paper [21]addressed state-feedback stabilization, but the designedcontroller is only robust against the stochastic inversedynamics with stringent stability margin.

This paper investigates the state-feedback stabi-lization for a class of stochastic high-order nonlinearsystems with SISS inverse dynamics. The main contri-butions are as follows:

(1) In this paper, we show that the condition onstochastic inverse dynamics can be relaxed tostochastic input-to-state stability (SISS) forsystem (1). By extending through adding apower integrator technique, choosing an appro-priate Lyapunov function and using the ideaof changing supply function, a smooth state-feedback controller is achieved to ensure thatthe equilibrium at the origin of the closed-loopsystem is globally asymptotically stable in prob-ability and the states can be regulated to theorigin almost surely.

(2) Two small-gain type conditions are indepen-dently used in [9] and [22], in Remark 3, wediscuss the relationship between these two condi-tions. What should be emphasized is that theconstruction of this counterexample in Remark3 is not easy.

This paper is organized as follows. Section IIprovides some mathematical preliminaries. A smoothstate-feedback controller is designed and analyzedin Section III, and is demonstrated by a simulationexample in Section IV. A concluding remark is givenin Section V.

II. PRELIMINARIES

The following notations will be used throughoutthe paper. R+ denotes the set of all nonnegative realnumbers and Rn denotes the real n-dimensional space.For a given vector or matrix X , XT denotes its trans-pose, Tr{X} is its trace when X is square, |x | denotesthe usual Euclidean norm of a vector x , and ‖X‖=(Tr{XT X}) 1

2 is the norm of a matrix X . Ci denotes theset of all functions with continuous i th partial deriva-tives. K denotes the set of all functions: R+ →R+,which are continuous, strictly increasing and vanishingat zero; K∞ denotes the set of all functions which areof class K and unbounded; KL denotes the set of allfunctions �(s, t): R+×R+ →R+, which are of classKfor each fixed t , and decrease to zero as t→∞ for eachfixed s.

Consider the following stochastic nonlinearsystem

dx= f (x)dt+gT (x)d�, x(0)=x0∈Rn, (2)

where x ∈Rn is the state of the system, � is anr -dimensional standard Wiener process defined onthe complete probability space (�,F, P). The Borelmeasurable functions f :Rn →Rn and g :Rn →Rr×n

are locally Lipschitz in x ∈Rn .The following definitions and lemmas will be used

throughout the paper.

Definition 1 ([23]). For any given V (x)∈C2, associ-ated with stochastic nonlinear system (2), the differen-tial operator L is defined as

LV=�V (x)

�xf (x)+1

2Tr

{g(x)

�2V (x)

�x2gT (x)

}. (3)

Definition 2 ([23]). For the stochastic nonlinearsystem (2) with f (0)=0, g(0)=0, the equilibriumx(t)=0 is globally asymptotically stable (GAS) inprobability if for any ε>0, there exists a class KLfunction �(·, ·) such that P{|x(t)|<�(|x0|, t)}≥1−ε

for any t≥0 and x0∈Rn \{0}.The following lemma gives a sufficient condition

on GAS in probability.

Lemma 1 ([23]). Consider the stochastic nonlinearsystem (2), if a C2 function V (x), class K∞ functions�(·), �(·), constants c1>0 and c2≥0, and a nonnegativefunction W (x) exist so that �(|x |)≤V (x)≤�(|x |),LV (x)≤−c1W (x)+c2, then

(a) For (2), there exists an almost surely unique solu-tion on [0,∞);


L. Liu and X.-J. Xie: State-Feedback Stabilization for Stochastic 209

(b) When c2=0, f (0)=0, g(0)=0, and W (x)=�(|x |), where �(·) is a class K function, thenthe equilibrium x=0 is GAS in probability andP{limt→∞ |x(t)|=0}=1.

The following lemmas will be used in the designof controller.

Lemma 2 ([24]). Let x1, . . . , xn, p be positive realnumbers, then

(x1+·· ·+xn)p ≤max{n p−1,1}(x p

1 +·· ·+x pn ).

Lemma 3 ([25]). For any positive real numbers m, nand any real-valued function �(x, y)>0, the followinginequality holds

|x |m |y|n ≤ m

m+n�(x, y)|x |m+n

+ n

m+n�−m

n (x, y)|y|m+n.

Lemma 4 ([26]). Let p∈R∗ and x , y be real-valuedfunctions. For a constant c>0, one has

|x p− y p| ≤ p|x− y|(x p−1+ y p−1)

≤ c|x− y||(x− y)p−1+ y p−1|.

III. CONTROLLER DESIGN ANDANALYSIS

3.1 Some assumptions and discussions

We need the following assumptions.

Assumption 1. For smooth functions fi (z, xi ) andgi (z, xi ) in (1), i=1, . . . ,n, there exist known nonneg-ative smooth functions �i,1(·), �i,2(·) so that

| fi (z, xi )| ≤ (|z|p+|x1|p+·· ·+|xi |p)�i,1(xi ), (4)

|gi (z, xi )| ≤ (|z| p+12 +|x1| p+1

2 +·· ·+|xi | p+12 )

·�i,2(xi ). (5)

Remark 1. Assumption 1 is similar to Assumption 2in [14] and Assumption 1 in [15], whose significanceand necessity is illustrated in these papers.

Assumption 2. z-subsystem in (1) is SISS, i.e. thereexist aC2 function V0(z) and classK∞ functions �1(·),�2(·), �0(·), �0(·) such that

�1(|z|) ≤ V0(z)≤�2(|z|),LV0(z) ≤ −�0(|z|)+�0(|x1|). (6)

As in the deterministic case, the function V0satisfying (6) is said to be a SISS-Lyapunov function,and (�0,�0) in (6) is called the SISS supply rate ofz-subsystem.

With this concept, we cite two small-gain typeconditions which are independently given in [9, 22], anddiscuss the relationship of these two conditions.

Lemma 5 ([22]). For the z-subsystem in (1) satisfying(6), if there is a class K∞ function �(·) such that

limsups→0+

�(s)

�0(s)<∞, limsup

s→∞�(s)

�0(s)<∞, (7)

then there exists a function �∈K∞ such that (�,�) is anew SISS supply rate of z-subsystem in (1).

Proof. From (7), it is easy to find a suitable constant�>0 such that for all s≥0, one has

�(s)≤��0(s). (8)

Choosing V (z)=�V0(z), by (3), (6) and (8), one gets

LV (z) = �LV0(z)

≤ −��0(|z|)+��0(|x1|)= −�(|z|)+��0(|x1|)−��0(|z|)+�(|z|)≤ −�(|z|)+��0(|x1|).

Setting �=��0, it follows that �∈K∞ and (�,�) is anew SISS supply rate of z-subsystem. �

We give another condition on SISS in [9].Assumption 3. For functions g0(z, x1) in (1) andV0(z) in (6), there exist known smooth nonnegativefunctions �z(·) and �0(·) such that |∇V0(z)|≤�z(|z|),‖g0(z, x1)‖≤�0(|z|), where ∇V0(z)= �V0(z)

�z .

Remark 2. Assumptions 2 and 3 are the generalassumption conditions and easy to satisfy. FromAssumptions 2, we see that z-subsystem is SISS withrespect to the input x1. In Assumptions 3, |∇V0(z)|≤�z(|z|) is a general assumption, ‖g0(z, x1)‖≤�0(|z|)is a constraint on the diffusion vector field of inversedynamics, which reflects that the diffusion vectorfield of inverse dynamics is confined by the dynamicsthemselves.

Lemma 6 ([9]). For the z-subsystem in (1) satisfying

(6), if limsups→0+�2z (s)�

20(s)

�0(s)<∞, Assumptions 3 hold,



and there is a class K∞ function �(s) such that

limsups→0+

�(s)

�0(s)<∞, (9)

∫ ∞

0[�(�−1

1 (s))]′e−∫ s0 [(�−11 ())]−1dds<∞, (10)

where �(s)≥0 and (s)>0 are continuous increasingfunctions defined on [0,∞) satisfying

�(s)�0(s)≥4�(s), (s)�0(s)≥2�2z (s)�

20(s), (11)

then there exist a nondecreasing positive function �(s)∈C1[0,∞) such that for any z∈Rm ,

�(V0(z))�0(|z|)≥2�′(V0(z))�2

z (|z|)�20(|z|)+4�(|z|), (12)

and a function �∈K∞ such that (�,�) is a new SISSsupply rate of z-subsystem in (1).

Remark 3. Since Lemma 5 and Lemma 6 are indepen-dently given in [9] and [22], this paper will discuss therelationship between (7) and (9)–(10).

Obviously, (7) is simpler and easier to verify

than (9)–(10). But if limsups→0+�2z (s)�

20(s)

�0(s)<∞ and

Assumption 3 hold, then (9)–(10) is strictly weakerthan (7). We prove this conclusion as follows:

Firstly, if (7) holds, one can choose a suitable posi-tive constant � so that �(s)≤��0(s), hence �(s) in (11)can be chosen as 4�, then for any satisfying (11), onehas ∫ ∞

0[�(�−1

1 (s))]′e−∫ s0 [(�−11 ())]−1dds=0. (13)

That is, (7) implies (9) and (10).However, its converse is not true by the following

counterexample:

dz=(−6z+x21)dt+z sin x1d�, (14)

where z and x1 are the system state and input, respec-tively. Choose the Lyapunov function V0(z)= 1

2 z2, then

LV0(z) = z(−6z+x21)+ 12 z

2 sin2 x1

≤ −5z2+ 12 x

41 .

Therefore, the functions �1, �0 and �0 in (6) can bechosen as �1(s)= 1

2 s2, �0(s)=5s2 and �0(s)= 1

2 s4.

Let �z(s)=s, �0(s)=s, then |g0(z, x1)|=|z sin x1|≤�0(|z|), | �V0(z)�z |≤�z(|z|), and limsups→0+

�2z (s)�

20(s)

�0(s)=

limsups→0+ 15s

2=0<∞. If we choose �(s)=s4, then

(9) holds, and 4�(s)�0(s)

= 45s

2,2�2

z (s)�20(s)

�0(s)≤ 2

5s2+1. Choose

the functions �(s)= 45 s

2, (s)= 25s

2+1, then (11) issatisfied. Through some simple calculations, one has∫ ∞

0[�(�−1

1 (s))]′e−∫ s0 [(�−11 ())]−1dds

= 8

5

∫ ∞

0

1

( 45s+1)54

ds<∞.

Hence �(s) satisfies (9) and (10). However, �(s)does not satisfy the second inequality of (7), i.e.limsups→∞

�(s)�0(s)

= limsups→∞ 15s

2=∞.

Remark 4. This Remark discusses the relationshipbetween these two conditions. We need to emphasizethat the construction of this counterexample is not easy.

3.2 Controller design

The objective of this paper is to design a smoothstate-feedback controller so that the equilibrium of theclosed-loop system is globally asymptotically stable inprobability and the states can be regulated to the origin.

Firstly, the following coordinate change is intro-duced

�1 = x1,

�i = xi −x∗i (xi−1),

(15)

where x∗i (xi−1)=−�i−1(xi−1)�i−1, i=2, . . . ,n, are

virtual smooth controllers with �i−1(·)≥0.With (15), Assumption 1 can be changed into the

following form.

Assumption 1. For smooth functions fi (z, xi ) andgi (z, xi ), i=1, . . . ,n, there are known nonnegativesmooth functions �i,1(·), �i,2(·) and �i,2(·) so that

| fi (z, xi )| ≤ (|z|p+|�1|p+·· ·+|�i |p)�i,1(xi ), (16)

|gi (z, xi )| ≤ (|z| p+12 +|�1|

p+12 +·· ·+|�i |

p+12 )

·�i,2(xi ), (17)

|gi (z, xi )|2 ≤ (|z|p+1+|�1|p+1+·· ·+|�i |p+1)

·�i,2(xi ). (18)

Proof. By (4), (15) and Lemma 2, one has

| fi (z, xi )|≤(|z|p+|x1|p+|x2|p+·· ·+|xi |p)�i,1(xi )



≤(|z|p+|�1|p+2p−1(|�2|p+|x∗2 |p)+·· ·

+2p−1(|�i |p+|x∗i |p))�i,1(xi )

=(|z|p+(2p−1�p1 +1)|�1|p+·· ·+(2p−1�p

i−1

+2p−1)|�i−1|p+2p−1|�i |p)�i,1(xi )≤(|z|p+|�1|p+·· ·+|�i |p)�i,1(xi ).

Similar to the proof of (16), by (5), (15) and Lemma 2,one can prove that (17), (18) hold. �

Next we provide the design procedure of thesmooth state-feedback controller.

Step 1: Introduce the first Lyapunov functionV1(x1)= 1

4k1�41, where k1>0 is a constant. With the

help of (1), (3), (15), Lemmas 2, 3 and Assumption 1,it can be verified that

LV1

=k1�31(x

p2 + f1)+ 3

2k1�

21Tr{gT1 g1}

=k1�31x

p2 +k1�

31 f1+ 3

2k1�

21|g1|2

≤k1�31x

p2 +k1|�1|3(|z|p+|�1|p)�1,1

+3

2k1|�1|2(|z|

p+12 +|�1|

p+12 )2�21,2

≤k1�31x

p2 +k1�1,1|z|p|�1|3+3k1�

21,2|z|p+1|�1|2

+(k1�1,1+3k1�21,2)�

p+31

≤k1�31x

p2 +( 1,1+ 1,2)|z|p+3+�1(x1)�

p+31 , (19)

where �1(x1)=k1�1,1+3k1�21,2 + 3p+3 (

p(p+3) 1,1

)p3 ·

kp+33

1 (1+ �21,1)p+36 + 2

p+3 (p+1

(p+3) 1,2)p+12 (3k1)

p+32 �p+3

1,2 ≥0, and 1,1, 1,2>0 are constants. Substituting the firstsmooth virtual controller

x∗2 (x1) = −�1(x1)�1,

�1(x1) =⎛⎜⎝c1,1+

√1+�21(x1)+

√1+�20(x1)

k1

⎞⎟⎠

1p

(20)

into (19) leads to

LV1 ≤ −c1,1�p+31 −�0(x1)�

p+31

+k1�31(x

p2 −x∗p

2 )+ 1|z|p+3, (21)

where c1,1>0, 1= 1,1+ 1,2, and �0(·) is a nonnegativesmooth function to be designed.

Step i(i=2, . . . ,n): Suppose that at step i−1,there exist a set of virtual controllers x∗

2 , . . . , x∗i defined

by (15) so that the (i−1)th Lyapunov function candi-date Vi−1(xi−1)= 1

4

∑i−1j=1 k j�

4j satisfies

LVi−1 ≤ −i−1∑j=1

ci−1, j�p+3j −�0(x1)�

p+31

+i−1∑j=1

j |z|p+3+ki−1�3i−1(x

pi −x∗p

i ), (22)

where ci−1, j , j , k j ( j =1, . . . , i−1) are positiveconstants. In the sequel, we will prove that (22) stillholds for the i th Lyapunov function candidate

Vi (xi )=Vi−1(xi−1)+ 14ki�

4i . (23)

By (1), (15), and Ito’s differential rule, one has

d�i =(x pi+1+ fi −

i−1∑k=1

�x∗i

�xk(x p

k+1+ fk)

−1

2

i−1∑l,m=1

�2x∗i

�xl�xmgTl gm

)dt

+(gi −

i−1∑k=1

�x∗i

�xkgk

)d�,

which together with (3), (22) and (23) imply that

LVi ≤ −i−1∑j=1

ci−1, j�p+3j −�0(x1)�

p+31 +

i−1∑j=1

j |z|p+3

+ki�3i x

pi+1+ki−1�

3i−1(x

pi −x∗p

i )

+ki�3i

(fi −

i−1∑k=1

�x∗i

�xk(x p

k+1+ fk)

−1

2

i−1∑l,m=1

�2x∗i

�xl�xmgTl gm

)

+3

2ki�

2i

∣∣∣∣∣gi −i−1∑k=1

�x∗i

�xkgk

∣∣∣∣∣2

. (24)

Using (15), Lemmas 2–4 and Assumption 1’, onegets

ki−1�3i−1(x

pi −x∗p

i )

≤cki−1|�i−1|3|�i ||(xi −x∗i )

p−1+x∗p−1i |



=cki−1|�i−1|3|�i |(|�i |p−1+|�i−1�i−1|p−1)

≤(bi,i−1,1+bi,i,1)�p+3i−1 +�i,1(xi−1)�

p+3i ,

ki�3i fi

≤ki |�i |3(|z|p+|�1|p+·· ·+|�i |p)�i,1≤ i,1|z|p+3+bi,1,2�

p+31 +·· ·

+bi,i−1,2�p+3i−1 +�i,2(xi )�

p+3i ,

−ki�3i

i−1∑k=1

�x∗i

�xk(x p

k+1+ fk)

≤i−1∑k=1

ki

√1+

(�x∗

i

�xk

)2

|�i |3((�k+1+x∗k+1)

p

+(|z|p+|�1|p+·· ·+|�k |p)�k,1)

=i−1∑k=1

ki

√1+

(�x∗

i

�xk

)2

|�i |3((|z|p+|�1|p+·· ·

+|�k−1|p)�k,1+(2p−1�pk + �k,1)|�k |p

+2p−1|�k+1|p)≤ i,2|z|p+3+bi,1,3�

p+31 +·· ·

+bi,i−1,3�p+3i−1 +�i,3(xi−1)�

p+3i ,

−1

2ki�

3i

i−1∑l,m=1

�2x∗i

�xl�xmgTl gm

≤ 1

2ki

i−1∑l,m=1

√1+�2i

√√√√1+(

�2x∗i

�xl�xm

)2

|�i |2

·(|z| p+12 +|�1|

p+12 +·· ·+|�l |

p+12 )

·(|z| p+12 +|�1|

p+12 +·· ·+|�m | p+1

2 )�l,2�m,2

≤ 1

2ki

i−1∑l,m=1

√1+�2i

√√√√1+(

�2x∗i

�xl�xm

)2

|�i |2

·(|z|p+1+|�1|p+1+·· ·+|�l |p+1+·· ·+|�m |p+1)�lm,2

≤ i,3|z|p+3+bi,1,4�p+31 +·· ·+bi,i−1,4�

p+3i−1

+�i,4(xi )�p+3i ,

3

2ki�

2i

∣∣∣∣∣gi −i−1∑k=1

�x∗i

�xkgk

∣∣∣∣∣2

≤3ki�2i

(|gi |2+(i−1)

i−1∑k=1

|�x∗i

�xkgk |2

)

≤3ki�2i

((|z|p+1+|�1|p+1+·· ·+|�i |p+1)�i,2

+(i−1)i−1∑k=1

(�x∗

i

�xk

)2

(|z|p+1+|�1|p+1+·· ·

+|�k |p+1)�k,2

)

≤ i,4|z|p+3+bi,1,5�p+31 +·· ·+bi,i−1,5�

p+3i−1

+�i,5(xi )�p+3i , (25)

where �lm,2 is a smooth function, bi,i−1,1, bi,i,1,bi,1,2, . . . ,bi,i−1,2, bi,1,3, . . . ,bi,i−1,3, bi,1,4, . . . ,bi,i−1,4,bi,1,5, . . . ,bi,i−1,5, i,1, i,2, i,3, i,4 are some designedpositive constants with

ci,1 = ci−1,1−bi,1,2−·· ·−bi,1,5>0,

...

ci,i−2 = ci−1,i−2−bi,i−2,2−·· ·−bi,i−2,5>0, (26)

ci,i−1 = ci−1,i−1−bi,i,1−bi,i−1,1 . . .−bi,i−1,5>0.

Substituting (25)–(26) into (24), and choosing the i thsmooth virtual controller

x∗i+1(xi ) = −�i (xi )�i ,

�i (xi ) =⎛⎝ci,i +

√1+�2i (xi )

ki

⎞⎠

1p

,

ci,i>0, (27)

one leads to

LVi ≤ −i∑

j=1ci, j�

p+3j −�0(x1)�

p+31 +

i∑j=1

j |z|p+3

+ki�3i (x

pi+1−x∗p

i+1), (28)

where �i (xi )=�i,1(xi−1)+�i,2(xi )+�i,3(xi−1)+�i,4(xi )+�i,5(xi ), i = i,1+ i,2+ i,3+ i,4.

When i=n, by choosing the smooth actual controllaw

u = x∗n+1(x)=−�n(x)�n,

�n(x) =(cn,n+√1+�2n(x)

kn

) 1p

,

cn,n>0, (29)



one has

LVn ≤−n∑

i=1cn,i�

p+3i −�0(x1)�

p+31 +�(|z|), (30)

where

Vn(x)= 1

4

n∑i=1

ki�4i ,�(|z|)=

n∑i=1

i |z|p+3 (31)

and cn,i , i=1, . . . ,n, are positive real numbers.

Remark 5. For system (1), in the design procedureof controller, we only give the existence of �i,1(·),�i,2(·), �i,3(·), �i,4(·) and �i,5(·)(i=2, . . . ,n) by usingLemmas 2-4 rather than their explicit definitions. Whilefor a detailed example, by appropriately choosingdesign parameters, �i,1(·), �i,2(·), �i,3(·), �i,4(·) and�i,5(·)(i=2, . . . ,n) can be concretely obtained, so thestate-feedback controller (29) can be implemented, seeSection IV for the details. From (20), (27) and (29), itis easy to see that x∗

i is positive smooth function withrespect to xi−1, i=2, . . . ,n+1.

Remark 6. The high-order of system (1) (p≥1) andthe appearance of the Hessian term in (3) will inevitablycause too large control effort. By introducing k1, . . . ,knappropriately, the control effort can be reduced effec-tively.

3.3 Main results

We state the main results in this paper.

Theorem 1. If Assumptions 1–3, (9)–(11), limsups→0+�2z (s)�

20(s)

�0(s)<∞, and

limsups→0+

�0(s)

s p+3<∞, (32)

hold for system (1), where �(s)=�(s), thus

(i) The closed-loop system consisting of (1), (15),(20), (27) and (29) has an almost surely uniquesolution on [0,∞);

(ii) The equilibrium at the origin of the closed-loopsystem is globally asymptotically stable in prob-ability and P{limt→∞(|z(t)|+∑n

i=1 |xi (t)|)=0}=1.

Proof. Supposing that �(·) is the function defined as inLemma 6, and setting

U (z)=∫ V0(z)

0�()d, (33)

by (28) in [9] and Assumptions 2, 3, one has

LU (z) = �(V0(z))LV0(z)

+1

2�′(V0(z))‖∇V T

0 (z)g0(z, x1)‖2

≤ �(V0(z))(−�0(|z|)+�0(|x1|))

+1

2�′(V0(z))�2

z (|z|)�20(|z|)

≤ �(�(|x1|))�0(|x1|)−1

2�(V0(z))�0(|z|)

+1

2�′(V0(z))�2

z (|z|)�20(|z|), (34)

where �(·)=�2(�−10 (2�0(·)))∈K∞.

Consider the Lyapunov function candidate for theentire system

W (z, x)=U (z)+Vn(x), (35)

where Vn(x) is defined by (31). Then, it follows from(30) and (34) that

LW (z, x) ≤ −n∑

i=1cn,i�

p+3i −�0(x1)x

p+31 +�(|z|)

+�(�(|x1|))�0(|x1|)−1

2�(V0(z))�0(|z|)

+1

2�′(V0(z))�2

z (|z|)�20(|z|). (36)

By Lemma 6, we have

1

4�(V0(z))�0(|z|)≥

1

2�′(V0(z))�2

z (|z|)�20(|z|)+�(|z|). (37)

From (32), one can construct a smooth function �0(·)such that

�0(x1)xp+31 ≥�(�(|x1|))�0(|x1|). (38)

Using (36)–(38), one gets

LW (z, x)≤−1

4�(0)�0(|z|)−

n∑i=1

cn,i�p+3i . (39)

From (31), (33), (35), (39), �0(·)>0 and Lemma 1, itfollows that the equilibrium at the origin of the closed-loop system is GAS in probability, and conclusion(i) and P{limt→∞(|z(t)|+∑n

i=1 |�i (t)|)=0}=1 hold,from which and (15), it is easy to recursively prove thatxi (t)→0 a.s. as t→∞, i=1, . . . ,n. �



IV. A SIMULATION EXAMPLE

Consider the following system

dz = (−2z3+ 12 x

31)dt+ 1

3 sin(zx1)d�,

dx1 = (x32 + 310 z

2x1)dt+ 15 x1 sin zd�,

dx2 = (u3+ 310 z

2x1 sin x2)dt

+ 110 (z

2+x21 sin x2)d�, (40)

where f1(z, x1)= 310 z

2x1, f2(z, x2)= 310 z

2x1 sin x2,g1(z, x1)= 1

5 x1 sin z, g2(z, x2)= 110 (z

2+x21 sin x2).Obviously, Assumption 1 is satisfied with �11= 1

5 ,

�21= 15

√1+x22 , �12= �22= 1

10 .Now, we apply the above design procedure to (40).

Introducing �1= x1 and choosing V1(x1)= 14k1�

41, it is

easy to deduce from (40) that

LV1≤−c1�61−�0�

61+0.2z6+k1�

31(x

32−x∗3

2 ), (41)

with x∗2 =−b1x1, b1=(

c1k1

+0.1k1+ k212500 +0.23+

�0k1

)1/3. Defining �2= x2−x∗2 = x2+b1x1, by Lemmas

2–4, one obtains

k1�31(x

32 −x∗3

2 )≤(d1+d2)�61+�2,1�

62,

k2�32 · 3

10z2x1 sin x2≤ 1

10z6+ 1

10�61+�2,2�

22,

k2b1�32 ·(x32 + 3

10z2x1

)

≤ 1

10z6+d3�

61+�2,3�

62,

3

2k2�

22

∣∣∣∣ 110 (x21 sin x2+z2)+ b15x1 sin z

∣∣∣∣2

≤ 1

10z6+d4�

61+�2,4�

62. (42)

where �2,1= 9k214d1

+ 16 (

56d2

)5(9k1b212 )6, �2,2= k22

5 (1+x22),

�2,3=0.1b21k22+0.25d−1

3 (b1k2(4b31+0.2))2+4b1k2,

�2,4=400(0.02k2(1+b21))3+4d−2

4 (0.02k2(1+b21))3.

Choosing V2(x1, x2)=V1(x1)+ 14k2�

42, with (41) and

(42), a direct calculation leads to

LV2 ≤ −(c1−0.1−d1−d2−d3−d4)�61

−�0�61+0.5z6+k2�

32u

3+�2(x2)�62, (43)

where �(s)=0.5s6, �2(x2)=�2,1+�2,2+�2,3+�2,4,and k1,k2,d1,d2,d3,d4 are positive design constants.

By �(s)=0.5s6, for z-subsystem of (40),with the choice of function V0(z)= 1

4 z4, one can

0 1 2 3 4 5 6 7

0

5

10

Time (Sec)

Sys

tem

sta

tes

0 1 2 3 4 5 6 7

0

2

4

6

8

10

Time (Sec)

Con

trol

inpu

t

u

zx1

x2

Fig. 1. The response of the closed-loop system (40) and (44).

verify LV0(z)≤−1.5z6+0.26x61 , |∇V0(z)|≤|z|3,|g0(z, x1)|≤ 1

3 , Assumptions 2 and 3 are satisfied with�0(s)=1.5s6, �0(s)=0.26s6, �z(s)=s3, �0(s)= 1

3 ,respectively. By some simple calculations, (9)–(12),(32) and (38) hold with �= 4

3 , �0= 720 .

In simulation, we choose c1=2, c2=d1=d4=0.1, d2=1.5, d3=0.2, k1=1/30, k2=0.001, then b1=4.1356, and the control law

u=−(4370.4+0.1√1+x42)

13 (4.1356x1+x2). (44)

We choose the initial values z(0)=6.5, x1(0)=1.5 and x2(0)=−6.2, the sampling period=0.01, Fig. 1gives the response of the closed-loop system (40) and(44), which demonstrates the effectiveness of the controlscheme.

V. A CONCLUDING REMARK

Some issues under current investigation are: howto further weaken SISS to SiISS introduced in [22]for stochastic high-order nonlinear systems? Anotheris when z �=0 and satisfies Assumptions 2 and 3, forsystem (1) under the following assumption:

| fi (z, xi )| ≤ (|z|1 +|x1|1 +·· ·+|xi |1)�i,1(xi ),|gi (z, xi )| ≤ (|z|2 +|x1|2 +·· ·+|xi |2)�i,2(xi ),

how to find sets �1 and �2 and design controller tomaintain the desired control performance as in Theorem1 for any 1∈�1 and 2∈�2; The third is to find apractical example for System (1).



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Liang Liu is a doctor candidateat Qufu Normal University. Hiscurrent research interests includedecentralized adaptive control ofcomplex systems and stochasticnonlinear control.

Xue-Jun Xie is a professor atQufu Normal University. Hiscurrent research interests includestochastic nonlinear controlsystems and adaptive control.


state-feedback stabilization for stochastic high-order nonlinear systems with siss inverse dynamics

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