Cooperative control of multiple stochastic high-order nonlinear systems

Automatica 82 (2017) 218–225

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Brief paper

Cooperative control of multiple stochastic high-order nonlinear systems✩ Wuquan Li a , Lu Liu b , Gang Feng b a

School of Mathematics and Statistics Science, Ludong University, Yantai, 264025, China

b

Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong

article

info

Article history: Received 15 August 2016 Received in revised form 18 January 2017 Accepted 18 April 2017

Keywords: Cooperative control Directed network topology Stochastic high-order nonlinear systems

abstract Distributed cooperative control of multiple stochastic high-order nonlinear systems has not been addressed in literature. This paper presents an approach to design of distributed cooperative controllers for multiple stochastic high-order nonlinear systems under directed leader–followers type network topology via the so-called distributed integrator backstepping method. By using the algebraic graph theory and stochastic analysis method, it is shown that the output tracking errors between the followers and the leader can be tuned arbitrarily small while all the states of the closed-loop system remain bounded in probability. Finally, the effectiveness of the proposed control approach is illustrated on a stochastic underactuated mechanical system. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction This paper considers distributed cooperative control of a network composed of N stochastic high-order nonlinear systems as followers and one leader. The followers’ dynamics are described as follows: pij

dxij = (xi,j+1 + fij (¯xij ))dt + gij (¯xij )dω, pi,n

dxi,ni = (ui

i

j = 1, . . . , ni − 1,

+ fi,ni (¯xi,ni ))dt + gi,ni (¯xi,ni )dω,

yi = xi1 ,

(1)

where x¯ ij = (xi1 , . . . , xij )T ∈ Rj , ui ∈ R, yi ∈ R are the state, input, output of the ith follower, respectively, i = 1, . . . , N. pij ∈ ≥1

Rodd = {q ∈ R : q ≥ 1 and q is a ratio of odd integers} is called the ‘‘high-order’’ of system (1). ω is an m-dimensional independent standard Wiener process defined on the complete probability space (Ω , F , Ft , P ) with a filtration Ft satisfying the typical

✩ This work was supported by National Natural Science Foundation of China under Grant (No. 61573179), the Young Taishan Scholars Program of Shandong Province of China under Grant (No. tsqn20161043), and grants from the Research Grants Council of the Hong Kong Special Administrative Region of China (Nos. CityU/11203714 and CityU/11213415). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Xiaobo Tan under the direction of Editor Miroslav Krstic. E-mail addresses: [email protected] (W. Li), [email protected] (L. Liu), [email protected] (G. Feng).

http://dx.doi.org/10.1016/j.automatica.2017.04.052 0005-1098/© 2017 Elsevier Ltd. All rights reserved.

conditions. Specifically, Ft is increasing and right continuous while F0 contains all P-null sets. The functions fij (¯xij ) : Rj → R, gij (¯xij ) : Rj → R1×m , i = 1, . . . , N, j = 1, . . . , ni , are known C 1 functions. The leader’s output is defined as y0 (t ) ∈ R. For the single agent case of system (1) (i.e. N = 1), there are fruitful results based on basic stability theory of stochastic control systems (Khasminskii, 2012; Kushner, 1967). For example, when p1j = 1 for all j = 1, . . . , ni , Deng and Krstić (1999), Deng, Krstić, and Williams (2001), Krstić and Deng (1998) and Pan and Basar (1999) respectively introduce two kinds of Lyapunov functions for controller design: quadratic Lyapunov functions multiplied by different weighting functions and quartic Lyapunov functions. Subsequently, these design techniques are further developed by Liu, Jiang, and Zhang (2008); Liu, Zhang, and Jiang (2007) and Wu, Xie, and Zhang (2007). When p1j > 1, j = 1, . . . , ni , Li, Jing, and Zhang (2011) and Xie and Tian (2007) address the state-feedback stabilization problem and Li, Xie, and Zhang (2011) study the output-feedback controller design by using homogeneous theory. For linear multi-agent systems with stochastic noise (i.e. N > 1), Li, Li, Xie, and Zhang (2016), Li and Zhang (2009) and Ma, Li, and Zhang (2010) solve the cooperative control problems with various systems structure. When it turns to the cooperative control of stochastic nonlinear multi-agent system (1), there are only a few results with pij = 1 for all i = 1, . . . , N, j = 1, . . . , ni in (1). Specifically, Song, Cao, and Yu (2010) present a pinning control and achieve leader-following consensus for multi-agent systems described by nonlinear second-order dynamics. Meng, Lin, and Ren (2013) study the distributed robust cooperative tracking problem

W. Li et al. / Automatica 82 (2017) 218–225

219

for multiple non-identical second-order nonlinear systems with bounded external disturbances. Zhang and Lewis (2012) investigate the cooperative tracking control problem of higher-order nonlinear systems with Brunovsky form. Li and Zhang (2014) focus on the stochastic multi-agent systems with inherent nonlinear drift and diffusion terms. However, when some pij ’s satisfy pij > 1 and stochastic noise exists in system (1), to the best of authors’ knowledge, there is no result reported in open literature. The following stochastic underactuated mechanical system gives a practical example of such stochastic high-order nonlinear system (1). Example 1. The stochastic underactuated mechanical system (Xie & Duan, 2010), see Fig. 1, is described by:

θ¨ =

g l

x¨ = −

sin θ + k m1

x−

ks m2 l ks m1

cooperative control of multiple stochastic mechanical systems. Section 5 includes some concluding remarks.

(x − l sin θ )3 cos θ ,

(x − l sin θ )3 +

v m1

.

(2)

The following coordinate changes

 x4 =

3

x2 = x˙ 1 , ks m2 l

x3 = (x − l sin θ )

 x˙ 3 ,

u=

3

ks cos x1 m31 m2 l

3



ks m2 l

2

cos θ ,

v,

transform the mechanical system (1) to dx1 = x2 dt , dx2 = x33 dt + f2 (x1 , x2 )dt , dx3 = x4 dt , dx4 = udt + f4 (x1 , x2 , x3 , x4 )dt + g4 (x1 , x2 , x3 , x4 )dω, y = x1 ,

2. Preliminaries 2.1. Graph theory

 x1 = θ ,

Fig. 1. Stochastic underactuated mechanical system.

(3)

where f2 (x1 , x2 ), f4 (x1 , x2 , x3 , x4 ) and g4 (x1 , x2 , x3 , x4 ) are C 1 functions whose concrete form can be found in Xie and Duan (2010). Obviously, (3) is a special form of the stochastic high-order nonlinear system (1). We will use this example for simulation in Section 4. In this paper, the cooperative control problem of system (1) is investigated under a directed graph topology. By using the algebra graph theory and stochastic analysis method, distributed controllers are designed to ensure that the tracking error converges to an arbitrarily small neighborhood of zero. The main challenges of this work are twofold: (1) This paper is the first result on cooperative control of system (1). Different from the traditional results for a single stochastic high-order nonlinear system (Li, Jing et al., 2011; Li, Xie et al., 2011; Xie & Tian, 2007), distributed cooperative control for system (1) needs to consider the interactions among agents, which makes the controller design and stability analysis of the closed-loop systems much more difficult. New design tools and stability analysis technique should be developed. (2) Since some pij ’s are allowed to satisfy pij > 1 in system (1), the system considered in this paper is more general than those in Meng et al. (2013), Song et al. (2010) and Zhang and Lewis (2012). Due to the effect of pij ’s, the Jacobian linearizations are neither controllable nor feedback linearizable, thus the existing design tools are hardly applicable and some new design technique should be developed. The remainder of this paper is organized as follows. Section 2 is on preliminaries. Section 3 presents controllers design and stability analysis. Section 4 applies the theoretical results to

For a given vector or matrix X , X T denotes its transpose. Tr{X } denotes its trace when X is square, and |X | is the Euclidean norm of a vector X . Let G = (V , E , A) be the followers’ weighted digraph of order N with the set of nodes V = {1, 2, . . . , N }, set of arcs E ⊂ V × V , and a weighted adjacency matrix A = (aij )N ×N with nonnegative elements. The set of neighbors of vertex i is denoted by Ni = {j ∈ V : j can directly send information to agent i, i ̸= j}. aij > 0 if node j is a neighbor of node i and aij = 0 otherwise. Node j is reachable from node i if there is at least one path from i to j. The diagonal matrix D = diag(κ1 , κ2 , . . . , κN ) is the degree matrix, whose diagonal elements κi = j∈Ni aij . The Laplacian of a weighted digraph G is defined as L = D − A. In this paper, we consider a network consisting of N agents and one leader (labeled by 0) which is described by a graph G¯ = (V¯ , E¯ ), where V¯ = {0, 1, 2, . . . , N }, set of arcs E¯ ⊂ V¯ × V¯ . If (0, i) ∈ E¯ , then 0 ∈ Ni . A diagonal matrix B = diag(b1 , b2 , . . . , bN ) is the leader adjacency matrix associated with G¯ , where bi > 0 if node 0 is a neighbor of node i, and bi = 0, otherwise. Lemma 1 (Hu & Hong, 2007). The matrix H = L + B is positive stable if and only if node 0 is globally reachable in G¯ . 2.2. Stochastic analysis and inequalities Definition 1 (Krstić & Deng, 1998). A stochastic process x(t ) is said to be bounded in probability if |x(t )| is bounded in probability uniformly in t, i.e., lim sup P {|x(t )| > c } = 0.

c →∞ t >t 0

Lemma 2 (Lyapunov Inequality Mao & Yuan, 2006). For a random variable ξ defined on the probability space (Ω , F , Ft , P ) such that E {|ξ |t } < ∞ (t > 0), the following inequality holds:

{E {|ξ |s }}1/s ≤ {E {|ξ |t }}1/t , where 0 < s ≤ t. Lemma 3 (Krstić & Deng, 1998). Let V ∈ C 2,1 (Rn × R+ ; R+ ) and τ1 , τ2 be bounded stopping times such that 0 ≤ τ1 ≤ τ2 a.s. If V (x, t ) and LV (x, t ) are bounded on t ∈ [τ1 , τ2 ] a.s., then E [V (x, τ2 ) − V (x, τ1 )] = E



τ2

τ1

LV (x, t )dt .

220

W. Li et al. / Automatica 82 (2017) 218–225

Lemma 4 (Qian & Lin, 2001). Let x, y be real variables. For any positive integers m and n, and positive real numbers a and b, the following inequality holds: axm yn ≤ b|x|m+n +

n



m+n

m+n

−m/n

m

· a(m+n)/n b−m/n |y|m+n .

Lemma 5 (Polendo & Qian, 2005). For all x, y ∈ R and any positive real number p, the following inequality holds:

|xp − yp | ≤ p|x − y|(xp−1 + yp−1 ) ≤ c |x − y| |(x − y)p−1 + yp−1 |. 3. Distributed controllers design and stability analysis In this section, we focus on the distributed controllers design and stability analysis for system (1). To proceed further, we need the following assumptions. Assumption 1. p11 = p21 = · · · = pN1 = 1. Assumption 2. The leader is globally reachable in G¯ . Assumption 3. The leader’s output y0 (t ) ∈ R and y˙ 0 (t ) are bounded, and the bound is only available to a subgroup of followers whose neighbors contain the leader. Remark 1. It can be observed from (3) that stochastic underactuated mechanical systems is a special class of stochastic highorder nonlinear systems. Consider cooperative control of multiple stochastic underactuated mechanical systems (3) with N agents. It can be easily deduced that pi1 = pi3 = pi4 = 1 and pi2 = 3, i = 1, . . . , N. Therefore, Assumption 1 is obviously satisfied. The following definition describes the distributed practical output tracking problem studied in this paper. Definition 2. The distributed practical output tracking problem for system (1) is solvable if for any given ε > 0, there exists a set of distributed control laws such that: (a) all the states of the closed-loop system are bounded in probability; (b) for any initial value x(t0 ), there is a finite-time T (x(t0 ), ε) such that E |yi (t ) − y0 (t )|p0 +3 < ε,

∀t > T (x(t0 ), ε), i = 1, . . . , N ,

where p0 = max{pij , i = 1, . . . , N ; j = 1, . . . , ni }. Remark 2. As demonstrated by deterministic system case (Lin & Pongvuthithum, 2003), system (1) has uncontrollable mode, which makes it impossible to achieve asymptotical output tracking by any continuously differentiable controllers. Therefore, we consider practical output tracking rather than asymptotical output tracking in this paper. Remark 3. In Definition 2, the (p0 + 3)th moment bounded tracking is considered, which is a stronger result than the second-order moment boundedness and the first-order moment boundedness. In fact, by Lyapunov inequality in Lemma 2, one obtains E |yi (t ) − y0 (t )| ≤ {E |yi (t ) − y0 (t )|2 }1/2

≤ {E |yi (t ) − y0 (t )|p0 +3 }1/(p0 +3) ,

from which, it follows that the (p0 + 3)th moment boundedness implies the second-order moment boundedness in probability and boundedness in probability. Hence, the (p0 + 3)th moment bounded tracking considered in this paper is more general.

Remark 4. The systems studied in Li, Jing et al. (2011), Lin and Pongvuthithum (2003), Li, Xie et al. (2011), Polendo and Qian (2005), Qian and Lin (2001), Xie and Duan (2010) and Xie and Tian (2007) are single-agent high-order nonlinear systems (N = 1) whose control problems can be handled by the method of adding a power integrator since these systems are single-input and single output. In contrast, for multiple stochastic high-order nonlinear systems (1) (N > 1), the closed-loop system is multiinput and multi-output. Due to the communication constraint which is described by a directed communication graph, we cannot use the full information of the system for feedback control. The control design in this paper needs to consider the interactions among agents, coupling terms in dynamics, and the capability of information exchange among agents and so on, which makes the controller design and performance analysis of the closed-loop system much more difficult than those in Li, Jing et al. (2011), Lin and Pongvuthithum (2003), Li, Xie et al. (2011), Polendo and Qian (2005), Qian and Lin (2001), Xie and Duan (2010) and Xie and Tian (2007). Remark 5. Compared with Meng et al. (2013) and Song et al. (2010), the contributions of this paper include: (i) We consider nonlinear multi-agent systems (1) with arbitrary order, which includes the second-order dynamics in Meng et al. (2013) and Song et al. (2010) as a special case. (ii) We consider a stochastic system model, which is more practical than the deterministic models considered in Meng et al. (2013) and Song et al. (2010). (iii) For the nonlinear functions, we only need local Lipschitz condition, which is more general than the global Lipschitz condition required in Song et al. (2010). (iv) In Meng et al. (2013) and Song et al. (2010), all the pij ’s are required to be 1, which is more restrictive than Assumption 1 in this paper. Therefore, the system model considered in this paper is more general than that in Meng et al. (2013) and Song et al. (2010). The control objective in this paper is to design distributed controllers to solve the distributed practical output tracking problem for system (1). 3.1. Controller design In this subsection, a recursive design method for distributed output tracking control of system (1) is proposed. In what follows, for simplicity, we use fij and gij instead of fij (¯xij ) and gij (¯xij ) respectively by dropping x¯ ij for i = 1, . . . , N, j = 1, . . . , ni . Step I. In this step, we aim to design distributed virtual controllers x∗12 , . . . , x∗N2 . Define ξi1 = s=1 ais (yi − ys ) + bi (yi − y0 ). It follows from (1) and Assumption 1 that

N

dξi1 =

N 

ais (xi2 + fi1 − xs2 − fs1 ) + bi (xi2 + fi1

s=1 N    − y˙ 0 ) dt + ais (gi1 − gs1 ) + bi gi1 dω s=1



= di (xi2 + fi1 ) −

N 



ais (xs2 + fs1 ) − bi y˙ 0 dt

s =1

 +

di gi1 −

N 

 ais gs1

dω

s =1

= di (xi2 − x∗i2 )dt + di x∗i2 dt −

N  s=1

ais (xs2 − x∗s2 )dt

W. Li et al. / Automatica 82 (2017) 218–225

−

N 

N 



ais x∗s2 dt + di fi1 −

s=1

where βi12 is any positive constant and



ais fs1 − bi y˙ 0 dt

s =1

βˇ i12 =

N    + di gi1 − ais gs1 dω,

(4)

s =1

p0 + 1  p0 + 3 p0 + 3

2

p0 +3 1 , p0 +3 i1

ξ

Choosing Vi1 = p +2 ∗ xi2

LVi1 ≤ di ξi10

N  2  p0 + 2   ρi12 (Λ1 ) = di |gi1 |2 + 1 + ais |gs1 |2 + 1 . s=1

p +2

+ ξi10

Substituting (9)–(10) into (5) leads to

by (4) one can get



di fi1 − bi y˙ 0 −

N 

ais fs1



p +2 ∗ xi2

p +2

LVi1 ≤ di ξi10

− ξi10

N 

N 



di (xi2 − x∗i2 ) −

ais (xs2 − x∗s2 )



p +2

+ ξi10



di (xi2 − x∗i2 ) −

p +2

− ξi10

N 



ais (xs2 − x∗s2 )

+ βi1 ,

(11) (p +3)/(p +2)

s=1

N 

ρi1 (Λ1 )

s=1

s =1 N  2  p0 + 2 p0 +1   + ξi1 di gi1 − ais gs1 

2

p +3

ais x∗s2 + ξi10

s =1

s=1 p +2

−2/(p0 +1)

ais + bi .

s=1

+ ξi10

βi12

2

N

where di =

221

ais x∗s2 .

(5)

(p +3)/(p +1)

0 0 where ρi1 (Λ1 ) = βˇ i11 ρi110 + βˇ i12 ρi120 , βi1 = βi11 + βi12 . By Assumption 2 and Lemma 1, H = B + L is invertible. Define

s=1

By Assumption 3, there exists a positive constant c` such that

|y0 (t )| ≤ c` ,

|˙y0 (t )| ≤ c` .

(6)

From (6) one obtains

  N      ais fs1 (xs1 ) di fi1 (xi1 ) − bi y˙ 0 −   s=1 ≤ di |fi1 (xi1 )| + bi |˙y0 | +

N 

  ξ11 c11 + ρ11 (Λ1 )    ..   .. −1  .  .  = −H  .    x∗N2 ξN1 cN1 + ρN1 (Λ1 ) 

x∗12

p +3

≤ di 1 + fi12 (xi1 ) + bi c` +

p +2

− ξi10 ais

p +2

LVi1 ≤ −ci1 ξi10

ais |fs1 (xs1 )|

N 

(12)

With (12), (11) becomes

s=1







+ di ξi10

N 

(xi2 − x∗i2 ) 

ais (xs2 − x∗s2 ) + βi1 ,

(13)

s=1



1 + fs12 (xs1 )

where Λ1 = (x11 , . . . , xN1 ) and

where ci1 > 0 is a parameter to be designed, i = 1, . . . , N. Step II. In this step, we aim to design distributed virtual controllers x∗ij , 1 ≤ i ≤ N , 3 ≤ j ≤ ni . For i = 1, . . . , N, j = 2, . . . , ni , define

  N  ρi11 (Λ1 ) = di 1 + fi12 (xi1 ) + ais 1 + fs12 (xs1 )

ξij = xij − x∗ij ,

s=1

, ρi11 (Λ1 ),

(7) T

s=1

+ bi c` .

(8)

With (7)–(8), by Lemma 4 one has p +2

ξi10



di fi1 −

N 

ais fs1 − bi y˙ 0

(p +3)/(p0 +2) p0 +3

ξi1

,

(9)

where βi11 is any positive constant, βˇ i11 −1/(p0 +2)

3)βi11 ) . By Lemma 4 one also has

2

≤

2

p0 +2 p0 +3

p +1

ξi10



di |gi1 | +

N 

ais |gs1 |

− 2 −

(ξ ρ

)

(Λ1 ))p0 +1

(p +3)/(p0 +1) p +3 ≤ βi12 + βˇ i12 ρi120 (Λ1 )ξi10 ,

n 

∂ xsk

p

(xs,skk+1 + fsk ) − ∂ 2 x∗ij

j −1 

∂ xsk ∂ xpl s,p=1,s̸=p k,l=1 j −1 N  

∂ 2 x∗ij

∂ xsk ∂ xsl s=1, k,l=1,k̸=l 2 ∗ N j −1 1   ∂ xij

2 s=1 k=1 ∂ x2sk

Gij = gij − (10)

j−1 N   ∂ x∗ij s=1 k=1

−

s=1

ρ (

Fij = fij −

((p0 +

N  2    ais gs1  di gi1 −

p0 +1 i12 Λ1 i1 1/(p +1) 2 1 i1 i12 0

≤ξ =

=

s=1

p0 + 2

(15)

where

≤ βi11 + βˇ i11 ρi110

p +1

where αi,j−1 is a nonnegative smooth function to be determined later, 1 ≤ i ≤ N , 3 ≤ j ≤ ni . By Itoˆ ′ s formula, using (1) and (14) one has pij

s =1

ξi10

(14)

dξij = (xi,j+1 + Fij )dt + Gij dω,



≤ |ξi1 |p0 +2 ρi11 (Λ1 )

p0 + 2

x∗ij = −ξi,j−1 αi,j−1 ,

j −1 N   ∂ x∗ij s=1 k=1

∂ xsk

∂ x∗ij ∂ y0

y˙ 0

gsk gplT

gsk gslT ,

|gsk |2

gsk .

(16)

222

W. Li et al. / Automatica 82 (2017) 218–225 p0 −pij +4 1 , j=2 p0 −pij +4 ij

 ni

Choosing Vi = Vi1 +

ξ

with (13)–(15) one

obtains n i −1

p +3



+

p −pij +3

ξij 0

∗p

p

p +2

+ di ξi10

(xi2 − x∗i2 ) +

ni  p0 − pij + 3

ni 

p −pij +3

ξij 0

Fij

p0 −pi,n +3 pi,n

+ ξi,ni

i

ui

−ξ

i

p0 i1

N  +2

≤

 ais (xs2 − x∗s2 )

+

2

j=2

ξ

p0 −pij +3 ∗pij xi,j+1 ij



≤

|Gij |2

ni 

ξ

+ βi1 .

(17)

ρij4 =

By Lemmas 4–5 one has

ξ



βij2 +

ni 

pij xi,j+1

(

−

p0 −pij +3 

ξij

p0 − pij + 3

ρij5 =

(ξi,j+1 − ξij αij )pij − (−ξij αij )pij

p0 + 3

p +3 (δi,j+1,j + δ˜i,j+1,j )ξij 0 +

p0 + 3 pij + 1

p0 − pij + 3 2

n i −1



p +3

ρij1 ξi,j0+1 ,

βij1

−pij /(p0 −pij +3)

,

ρ

2 ij3

βij2

−(pij +1)/(p0 −pij +2)

(p0 +3)/(p0 −pij +2)

LVi ≤ −(ci1 − δi,2,1 )ξi10

.

p +2

− ξi10

N 



ais (xs2 − x∗s2 )

s =1

(18)

j =2

pij

p0 + 3



p0 + 3 p0 − pij + 3 (p +3)/pij

· λij 0

+

p0 + 3

)

δi,j+1,j 

1

p −1 p +3 0

· (λij αijij

ni −1

+

−(p0 −pij +3)/pij

p0 + 3 p0 + 2

δ˜i,j+1,j

+



p −pij +3 ∗pij xi,j+1

ξij 0

p0 −pi,n +3 pi,n

+ ξi,ni

i

ui

i

−(p0 +2)

p +3

,

where ρi,1,1 =

(xi2 − x∗i2 )

≤ di |ξi1 |p0 +2 |ξi2 | p +3

p +3

≤ δi21 ξi10

+

di 0

p0 + 3



p0 + 3 p0 + 2

δi21

−(p0 +2)

p +3

ξi20

,

(19)

where δi21 > 0 is any positive constant. It follows from the definition of Fij and Gij in (16) that there exist nonnegative smooth functions ρij2 (Λj ) and ρij3 (Λj ) such that

|Fij | ≤ ρij2 (Λj ),

|Gij | ≤ ρij3 (Λj ).

(20)

In fact, there are many choices for ρij2 and ρij3 , for instance, ρij2 =   1 + Fij2 and ρij3 = 1 + G2ij . By (20) and Lemma 4, the gradient term in (17) can be estimated as p0 −pij +3 Fij ij

≤

ni  j =2

(δi,j+1,j + δ˜i,j+1,j + ρi,j−1,1 (Λj−1 )

+ ρij4 (Λj ) + ρij5 (Λj ))ξij 0

By Lemma 4 one obtains p +2 di i10

ni  j =2

Λj = (ΛTi,j−1 , x1j , . . . , xNj )T , λij is a positive constant, δi,j+1,j and δ˜i,j+1,j are any positive constants.

j=2

(22)

j =2

ρij1 =

ξ

,

Substituting (18)–(19) and (21)–(22) into (17) yields

where

ni 



p +3

j =2

ξ

pij

p0 − pij + 2

·

  λij |ξij |p0 −pij +3 |ξi,j+1 | |ξi,j+1 |pi,j −1 + |ξij αij |pij −1

ni −1



p0 + 3

p0 + 3





j=2

≤



· ρij2

∗pij xi,j+1 )

n i −1



p +3

ρij5 (Λj )ξij 0

j =2

j =2

≤

|Gij |2

2 |ξij |p0 −pij +2 ρij3

(p0 +3)/(p0 −pij +3)

j=2 ni −1

p −pij +2

2

j=2

j=2

p0 −pij +3 ij

(21)

where βij1 and βij2 are any positive constants, and

n i −1

+

p0 −pij +2 ij

p +3

ρij4 (Λj )ξij 0

j =2

ξij 0

ni  p0 − pij + 3 j=2

s=1 ni  p0 − pij + 3

2

j =2

j =2

=

ni 

and the Hessian term in (17) can be estimated as

(xi,ijj+1 − xi,j+ij 1 )

j =2



βij1 +

j =2

LVi ≤ −ci1 ξi10

n i −1

ni 

≤

|ξij |p0 −pij +3 ρij2

p +3

di 0

p0 +3



δ

p0 +3 p0 +2 i21

+ βi ,

−(p0 +2)

(23)

, δi,ni +1,ni = δ˜ i,ni +1,ni = 0,

ni

βi = βi1 + j=2 (βij1 + βij2 ). For j = 2, . . . , ni − 1, choosing the following controllers,  x∗i,j+1 = − cij + δi,j+1,j + δ˜ i,j+1,j + ρi,j−1,1 (Λj−1 ) 1/pij + ρij4 (Λj ) + ρij5 (Λj ) ξij , αij ξij ,  ui = − ci,ni + ρi,ni −1,1 (Λni −1 ) + ρi,ni ,4 (Λni ) 1/pi,n i + ρi,ni ,5 (Λni ) ξi,ni , αi,ni ξi,ni , (24) and substituting them into (23) yields p +3

LVi ≤ −(ci1 − δi,2,1 )ξi10

−

ni 

p +3

cij ξij 0

j=2 p +2

− ξi10

N 



ais (xs2 − x∗s2 ) + βi ,

(25)

s=1

where cij , j = 2, . . . , ni , are parameters to be designed later and ξi,ni = xi,ni − x∗i,ni = xi,ni + ξi,ni −1 αi,ni −1 , Λni =

(ΛTni −1 , x1,ni , . . . , xN ,ni )T .

W. Li et al. / Automatica 82 (2017) 218–225

3.2. Stability analysis

+

N 

1

ci2 >

p0 + 3 s=1

p 0 + 3 s =1

V =

N

i=1

Vi , by (25) one has

i =1 p0 +2 i1

N 

i =1

=

N   ais (xs2 − xs2 ) + βi .

(27)

i =1

ais (xs2 − x∗s2 )

ξ

p0 + 3 i=1 s=1

β =



p +3

 ci1 − δi,2,1 −

 ci2 −

ni N  

+

1

N  N 

p0 + 3 i=1 s=1

N p0 + 2 

p0 + 3 s=1 N 

1

p0 + 3 s=1 p +3

cij ξij 0

+

i=1 j=3

p +3

ais ξs20

p +3

N 



 asi

βi .

N

1



p0 + 3 s=1



asi gi2 (ϵ)(p0 − pi2 + 4),

N 

 ci2 −

ni N  

N 

1

p0 + 3 s=1

 asi

δi2 ϵ (p0 +3)/(p0 −pi2 +4)

cij δij ϵ (p0 +3)/(p0 −pij +4) +

N 

βi .

(32)

i=1

Eectl V (ξ (tl )) ≤ ect0 EV (ξ (t0 )) + E

tl



ecs LV (ξ (s))ds

t0 tl

 + cE

,

ecs V (ξ (s))ds.

(33)

t0

− δij ϵ (p0 +3)/(p0 −pij +4) ,

p0 +3 (pij −1)/(p0 −pij +4) p0 +3 p0 −pij +4 pij −1 (pij −1)/(p0 −pij +4)

(

)

(29)

 ci1 − δi,2,1 −

N p0 + 2 

p0 + 3 s=1

e EV (ξ (t )) ≤ e ct

ct0

EV (ξ (t0 )) + E



t

ecs LV (ξ (s))ds t0

and



ecs V (ξ (s))ds, t0

which together with (31) implies

 ais

t

+ cE

gij (ϵ) = δij ϵ . With (26) and (29), (28) can be written as

i =1

p0 + 3 s=1



ais (p0 + 3),

Note liml→∞ ηl = ∞. Then, letting l → ∞, by (33) we have

≥ gij (ϵ)ξ

LV ≤ −

N p0 + 2 

(28)

i =1

p0 −pij +4 ij

N 

,

and tl = min{ηl , t } for all t ≥ t0 . Since |ξ | is bounded in the interval [t0 , tl ] a.s., V (ξ ) is bounded on [t0 , tl ] a.s. From (31), it can be obtained that LV is also bounded on [t0 , tl ] a.s. By Lemma 3 one has

p +3

ξi20

p +3

p −pij +4

ξij 0

ξ (t ) = (ξ11 , . . . , ξ1,n1 , . . . , ξN1 , . . . , ξN ,nN )T , ηl = inf{t : t ≥ t0 , |ξ | ≥ l}, ∀l > 0,

p +3

ξi10

ais

≤ ϵ + δij−1 ϵ −(pij −1)/(p0 −pij +4) ξij 0

where δij =

1

p0 − pij + 4 i=1 j=2

Noting that the local Lipschitz condition of the closed-loop system (1) and (24) holds, by (31) and Theorem 1 in Liu et al. (2007), the closed-loop system (1) and (24) has an almost surely unique strong solution on [0, ∞). Let

,

which yields

ξij 0

(30)

(31)

By Lemma 4, for any positive constant ϵ , one has p −pij +4

ni N  

i =1 j =3

ais ξi10

i =1

ξij 0

+

ci1 − δi,2,1 −

ci2 −

+

i =1

−

p +3



i=1

which, when being substituted into (27), yields

−

βi .



ξ

N N p0 + 2  

N 

N 

cij gij (ϵ)(p0 − pij + 4), i = 1, . . . , N , j = 3, . . . , ni ,

p +2 ais i10 s2

N 

+

i =1

ξi10

p0 + 3 i=1



s=1

LV ≤ −

p0 −pij +4

cij gij (ϵ)ξij

LV ≤ −cV + β,

i=1 s=1

≤

N 

1

c = min

∗

s=1

N  +2

N  N 

ni N  

with (26) and (30) one obtains

By (14) and Lemma 4 one obtains

i=1

cij δij ϵ (p0 +3)/(p0 −pij +4)

i=1 j=2

ξ

δi2 ϵ (p0 +3)/(p0 −pi2 +4)

asi

where

ni N N    p +3 p +3 LV ≤ − (ci1 − δi,2,1 )ξi10 − cij ξij 0

p

p0 + 3 s=1



Noting asi ,

Proof. Defining V =

ξi10

ci2 −

p −pi2 +4

gi2 (ϵ)ξi20

asi

i=1 j=3

(26)

N 

N 

1



i =1 j =3

ais ,

the distributed practical output tracking problem for system (1) is solvable.

N 



ni N  

+

−

ci2 > 0,

−

p0 + 3 s=1

i =1

Theorem 1. Consider a network of multiple stochastic high-order nonlinear systems (1). Assume Assumptions 1–3 hold. Under the distributed control law (24) with the design parameters chosen as N p0 + 2 

N 

N 

1

ci2 −

i=1

In this subsection, the stability analysis is presented for the closed-loop control system composed of (1), (12), (14) and (24). The main result is summarized in the following theorem.

ci1 > δi,2,1 −

N 

−

223



p +3

ξi10

β EV (ξ (t )) ≤ e−c (t −t0 ) EV (ξ (t0 )) + (1 − e−c (t −t0 ) ). c

(34)

224

W. Li et al. / Automatica 82 (2017) 218–225

Now we firstly show that for any given ε and initial value x(t0 ), there is a finite-time T (x(t0 ), ε) such that E |yi (t ) − y0 (t )|p0 +3 < ε,

∀t > T (x(t0 ), ε), i = 1, . . . , N .

Let ξ1 = (ξ11 , . . . , ξN1 ) . By (34) one has T

2 2 (p0 +3)/2 E |ξ1 |p0 +3 = E (ξ11 + · · · + ξN1 )

≤ N (p0 +1)/2 E (ξ110

p +3

p +3

+ · · · + ξN10 )

≤ N (p0 +1)/2 (p0 + 3)EV  ≤ N (p0 +1)/2 (p0 + 3) e−c (t −t0 ) EV (ξ (t0 )) +

β c

 (1 − e−c (t −t0 ) ) .

Fig. 2. The communication topology G¯ .

(35)

From the definition of ξs1 , s = 1, . . . , N, it can be obtained that

ξ1 =

N 

a1s (y1 − ys ) + b1 (y1 − y0 ), . . . ,

s =1 N 

aNs (yN − ys ) + bN (yN − y0 )

T

Remark 6. It can be observed from c > 0 and (37) that lim e−c (t −t0 ) EV (ξ (t0 )) = 0.

s=1

=

N 

N 

a1s (y1 − y0 ) −

s =1

t →∞

a1s (ys − y0 ) + b1 (y1 − y0 ), . . . ,

N

s=1

s=1

By the definition of c and β in (32), one obtains that the order of ϵ in p −1

s =1

N

By (40), ξ (t ) is bounded in probability. This together with Assumption 3 and (36) implies that yi = xi1 is bounded in probability, i = 1, . . . , N. Notice that ξi1 , ξi2 and xi1 are bounded in probability, from ξi2 = xi2 − x∗i2 and (12), one can conclude that xi2 is bounded in probability, i = 1, . . . , N. Similarly, one can prove that xij , i = 1, . . . , N, j = 3, . . . , ni , are bounded in probability. Therefore, all the states of the closed-loop system are bounded in probability. The proof is thus completed.

β is strict larger than that in c (i.e. p0p−0p+ij3+4 > p0 −ijpij +4 ). Therefore, ni noting βi = βi1 + j=2 (βij1 + βij2 ), by tuning ϵ , βi1 , βij1 and βij2 , i = 1, . . . , N , j = 1, . . . , ni , βc in (37) can be made arbitrarily small. From (37) one knows E |y − 1N y0 |p0 +3 can be made arbitrarily

T   aNs (yN − y0 ) − aNs (yN − y0 ) + bN (yN − y0 ) = (L + B)(y − 1N y0 ) = H (y − 1N y0 ).

(36)

By Assumption 2 and (35)–(36) one has

small. Therefore, the tracking error can be tuned to be arbitrarily small by using E |yi (t ) − y0 (t )|p0 +3 ≤ E |y − 1N y0 |p0 +3 . 4. A simulation example

E |y − 1N y0 |p0 +3



≤ N (p0 +1)/2 (p0 + 3)|H −1 |p0 +3 e−c (t −t0 ) EV (ξ (t0 )) (37)

Example 2. Consider cooperative control of a network composed of three followers and one leader. The followers’ dynamics are described by (3).

By the definition of c and β in (32), with (37), for any ε > 0 and x(t0 ), one can find a finite-time T (x(t0 ), ε) and choose suitable ϵ , βi1 , βij1 and βij2 , i = 1, . . . , N , j = 1, . . . , ni , such that

The communication topology of the concerned network is described in Fig. 2. The goal is to design distributed controllers such that the outputs of followers can track the leader’s output

E |yi (t ) − y0 (t )|p0 +3 ≤ E |y − 1N y0 |p0 +3 < ε,

y0 (t ) = 0.5 sin t .

 β + (1 − e−c (t −t0 ) ) . c

∀t > T (x(t0 ), ε), i = 1, . . . , N .

Define the output tracking errors as:

Next we show that all the states of the closed-loop system are bounded in probability. From (34) one has EV (ξ (t )) ≤ EV (ξ (t0 )) +

β c

.

(38)

Note that EV (ξ ) ≥



V (ξ )P (dw) ≥ inf V (ξ )P (|ξ | > c ). |ξ |>c

|ξ |>c

(39)

Then, by (38) and (39) we have P (|ξ | > c ) ≤

EV (ξ (t0 )) + inf V (ξ )

β c

,

|ξ |>c

which together with the definition of V (ξ ) gives lim sup P (|ξ | > c ) ≤ lim sup

c →∞ t >t 0

c →∞ t >t 0

EV (ξ (t0 )) + inf V (ξ )

|ξ |>c

β c

= 0.

(40)

ei = yi − y0 ,

i = 1, 2, 3.

Initial conditions are randomly set as x11 (0) = 3.2, x12 (0) = 1, x13 (0) = 0.2, x14 (0) = −1, x21 (0) = −0.1, x22 (0) = 0.1, x23 (0) = −0.3, x24 (0) = 0.2, x31 (0) = −2.5, x32 (0) = 0.2, x33 (0) = 1, x34 (0) = −0.3. The response of the closed-loop system is shown in Fig. 3. It can be observed from Fig. 3 that |ei | < ε = 0.2, ∀t > T (ε) = 10 s., i = 1, 2, 3. Thus, the effectiveness of the controller is clearly demonstrated. 5. Concluding remarks This paper studies the cooperative control problem of multiple stochastic high-order nonlinear systems under directed network topology. The designed distributed controllers can ensure that the output tracking errors between the followers and the leader can be tuned arbitrarily small while all the states of the closed-loop system remain bounded in probability. It is interesting to see whether or not these results can be extended to cooperative control of multiple stochastic nonlinear systems with time-varying topologies. Another challenging issue is distributed control with various communication constraints.

W. Li et al. / Automatica 82 (2017) 218–225

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Mao, X. R., & Yuan, C. G. (2006). Stochastic differential equations with Markovian switching. London: Imperial College Press. Meng, Z. Y., Lin, Z. L., & Ren, W. (2013). Robust cooperative tracking for multiple non-identical second-order nonlinear systems. Automatica, 49, 2363–2372. Pan, Z. G., & Basar, T. (1999). Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion. SIAM Journal on Control and Optimization, 37, 957–995. Polendo, J., & Qian, C. (2005). A generalized framework for global output feedback stabilization of nonlinear systems. In Proceedings of the 44th IEEE conference on decision and control (pp. 2646–2651). Qian, C., & Lin, W. (2001). A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Transactions on Automatic Control, 46(7), 1061–1079. Song, Q., Cao, J. D., & Yu, W. W. (2010). Second-order leader-following consensus of nonlinear multi-agent systems via pinning control. Systems & Control Letters, 59, 553–562. Wu, Z. J., Xie, X. J., & Zhang, S. Y. (2007). Adaptive backstepping controller design using stochastic small-gain theorem. Automatica, 43, 608–620. Xie, X. J., & Duan, N. (2010). Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system. IEEE Transactions on Automatic Control, 55, 1197–1202. Xie, X. J., & Tian, J. (2007). State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics. International Journal of Robust and Nonlinear Control, 17, 1343–1362. Zhang, H. W., & Lewis, F. (2012). Adaptive cooperative tracking control of higherorder nonlinear systems with unknown dynamics. Automatica, 48, 1432–1439.

Fig. 3. The response of tracking errors and controllers.

References Deng, H., & Krstić, M. (1999). Output-feedback stochastic nonlinear stabilization. IEEE Transactions on Automatic Control, 44, 328–333. Deng, H., Krstić, M., & Williams, R. J. (2001). Stabilization of stochastic nonlinear driven by noise of unknown covariance. IEEE Transactions on Automatic Control, 46, 1237–1253. Hu, J. P., & Hong, Y. G. (2007). Leader-following coordination of multi-agent systems with coupling time delays. Physica A, 374, 853–863. Khasminskii, R. (2012). Stochastic stability of differential equations (2nd ed.). London: Springer. Krstić, M., & Deng, H. (1998). Stabilization of uncertain nonlinear systems. New York: Springer. Kushner, H. J. (1967). Stochastic stability and control. New York: Academic. Li, W. Q., Jing, Y. W., & Zhang, S. Y. (2011). Adaptive state-feedback stabilization for a large class of high-order stochastic nonlinear systems. Automatica, 47, 819–828. Li, W. Q., Li, T., Xie, L. H., & Zhang, J. F. (2016). Necessary and sufficient conditions for bounded distributed mean square tracking of multi-agent systems with noises. International Journal of Robust and Nonlinear Control, 26, 631–645. Li, W. Q., Xie, X. J., & Zhang, S. Y. (2011). Output-feedback stabilization of stochastic high-order nonlinear systems under weaker conditions. SIAM Journal on Control and Optimization, 49, 1262–1282. Li, T., & Zhang, J. F. (2009). Mean square average-consensus under measurement noises and fixed topologies: necessary and sufficient conditions. Automatica, 45, 1929–1936. Li, W. Q., & Zhang, J. F. (2014). Distributed practical output tracking of high-order stochastic multi-agent systems with inherent nonlinear drift and diffusion terms. Automatica, 50, 3231–3238. Lin, W., & Pongvuthithum, R. (2003). Adaptive output tracking of inherently nonlinear systems with nonlinear parameterization. IEEE Transactions on Automatic Control, 48(10), 1737–1749. Liu, S. J., Jiang, Z. P., & Zhang, J. F. (2008). Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems. Automatica, 44, 1944–1957. Liu, S. J., Zhang, J. F., & Jiang, Z. P. (2007). Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems. Automatica, 43, 238–251. Ma, C. Q., Li, T., & Zhang, J. F. (2010). Consensus control for leader-following multi-agent systems with measurement noises. Journal of Systems Science and Complexity, 23, 35–49.

Wuquan Li received the Ph.D. degree in College of Information Science and Engineering, Northeastern University, China, in 2011. From 2012 to 2014, he carried out his postdoctoral research with Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China. Since January 2011, he has been with School of Mathematics and Statistics Science, Ludong University, where he is currently an associate professor. He is a Young Taishan Scholar. His research interests include stochastic nonlinear systems control and cooperative control of multi-agent systems. Lu Liu received her Ph.D. degree in 2008 in the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong. From 2009 to 2012, she was an Assistant Professor in The University of Tokyo, Japan, and then a Lecturer in The University of Nottingham, United Kingdom. She is currently an Assistant Professor in the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong. Her research interests are primarily in networked dynamical systems, control theory and applications, and biomedical devices. She received the Best Paper Award (Guan ZhaoZhi Award) in the 27th Chinese Control Conference in 2008. Gang Feng received the B.Eng and M.Eng. degrees in Automatic Control from Nanjing Aeronautical Institute, China in 1982 and in 1984 respectively, and the Ph.D. degree in Electrical Engineering from the University of Melbourne, Australia in 1992. He has been with City University of Hong Kong since 2000 where he is now Chair Professor of Mechatronic Engineering. He was lecturer/senior lecturer at School of Electrical Engineering, University of New South Wales, Australia, 1992–1999. He was awarded an Alexander von Humboldt Fellowship in 1997, and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007, and Changjiang chair professorship from Education Ministry of China in 2009. His current research interests include multiagent systems and control, intelligent systems and control, and networked systems and control. Prof. Feng is an IEEE Fellow, an associate editor of IEEE Trans. Fuzzy Systems and Journal of Systems Science and Complexity, and was an associate editor of IEEE Trans. Automatic Control, IEEE Trans. Systems, Man & Cybernetics, Part C, Mechatronics, and Journal of Control Theory and Applications.