Lyapunov formulation of the large-scale, ISS cyclic-small-gain theorem: The discrete-time case

Systems & Control Letters 61 (2012) 266–272

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Lyapunov formulation of the large-scale, ISS cyclic-small-gain theorem: The discrete-time case✩ Tengfei Liu a,∗ , David J. Hill b , Zhong-Ping Jiang c,d a

Research School of Engineering, The Australian National University, Canberra, ACT 0200, Australia

b

School of Electrical and Information Engineering, The University of Sydney and National ICT Australia, NSW 2006, Australia

c

Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, Six Metrotech Center, Brooklyn, NY 11201, USA

d

College of Engineering, Beijing University, PR China

article

info

Article history: Received 10 August 2010 Received in revised form 20 May 2011 Accepted 1 November 2011 Available online 10 December 2011

abstract This paper presents a Lyapunov formulation of the cyclic-small-gain theorem for dynamical networks composed of discrete-time input-to-state stable (ISS) subsystems. ISS-Lyapunov functions for dynamical networks satisfying the cyclic-small-gain condition are constructed from the ISS-Lyapunov functions of the subsystems. © 2011 Elsevier B.V. All rights reserved.

Keywords: Dynamical networks Discrete-time Input-to-state stability (ISS) Small-gain Lyapunov functions

1. Introduction The small-gain theorem has been widely recognized as one of the most important tools for stability analysis and control design in both linear and nonlinear systems. Truly nonlinear small-gain theorems, i.e., those allowing general nonlinear gains, started with [1,2] using input–output operator theory and [3] relying upon Sontag’s seminal work on input-to-state stability (ISS) [4,5] and its various equivalent characterizations by Sontag and Wang [6]. Early applications of nonlinear small-gain to control systems with saturation and dynamic uncertainties appear in [7,8], respectively. The discrete-time counterpart of continuous-time ISS and its small-gain theorem were developed later in [9,10]. Considering the relation between ISS and Lyapunov functions, the Lyapunov formulations of ISS small-gain theorems for continuoustime systems and discrete-time systems were reported in [11,12].

✩ This research was supported under the Australian Research Councils Discovery funding scheme (project number: FF0455875), in part by a seed grant from NYU and POLY, and by NSF grant DMS-0906659. ∗ Corresponding author. Tel.: +61 2 6125 8639; fax: +61 2 6125 8651. E-mail addresses: [email protected], [email protected] (T. Liu), [email protected] (D.J. Hill), [email protected] (Z.-P. Jiang).

0167-6911/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2011.11.002

The Lyapunov-based small-gain results for hybrid systems have also been developed in [13–15]. Recently, some interesting extensions of the nonlinear smallgain theorem were obtained for general networks composed of ISS subsystems in [16–21] for both continuous-time and discrete-time systems. In [18], the interconnections in large-scale systems are formulated with nonlinear matrices and the small-gain condition was obtained in matrix form. In [20,21], the dynamical network with ‘‘max’’-type interconnection was systematically studied and more general cyclic-small-gain criteria were developed for inputto-output stable (IOS) systems. The Lyapunov formulation of the matrix small-gain theorem for both continuous-time and discretetime systems was developed in [22]. The Lyapunov formulation of cyclic-small-gain for continuous-time dynamical networks was reported in our recent work [23]. It should be mentioned that the ‘‘max’’-type interconnection can also be represented with the notion of monotone aggregation functions in the recent work [22]. Refs. [24–26] show the equivalence of small-gain conditions and the existence of an asymptotically stable discretetime comparison system induced by the interconnection gains of dynamical networks. In particular, Refs. [24,25] proved a vector small-gain theorem based on the relation between the vector gains in the general dynamical networks and the asymptotic stability of discrete-time systems with ‘‘MAX’’-type system matrices.

T. Liu et al. / Systems & Control Letters 61 (2012) 266–272

In view of the critical importance of discrete-time system theory in computer-aided control engineering applications [27], in this paper, we take a lead in generalizing the results of [23] to discrete-time dynamical networks. Interestingly, as we will see later, new phenomena arise; see Remarks 1 and 3 below. Also, some new technical lemmas are needed to achieve this generalization. Just like the continuous-time case, we can systematically construct a total ISS-Lyapunov function on the basis of ISS-Lyapunov functions of individual subsystems. The rest of the paper is organized as follows. Section 2 gives some notations and definitions. Section 3 provides two ISS-Lyapunov formulations of the subsystems in the dynamical networks. Section 4 mainly studies the ISS-Lyapunov cyclic-smallgain for dynamical networks with subsystems formulated in dissipation form. In Section 5, we develop a counterpart of the results in Section 4 for dynamical networks with subsystems formulated in ‘‘gain margin’’ form. In Section 6, we employ an example to show the effectiveness of the main result. Section 7 presents some conclusions. The new technical lemmas are in Appendix. 2. Notations and definitions Throughout the paper, we use |x| to denote the Euclidean norm of x ∈ Rn and xT to denote the transpose of the vector x. We denote by Z+ the set of nonnegative integers. A function γ : R≥0 → R≥0 is positive definite if γ (s) > 0 for all s > 0 and γ (0) = 0. γ : R≥0 → R≥0 is a class K function if it is continuous, strictly increasing and γ (0) = 0; it is a class K∞ function if it is a class K function and also satisfies γ (s) → ∞ as s → ∞. For nonlinear functions γ1 and γ2 defined on R≥0 , inequality γ1 ≤ γ2 (or γ1 < γ2 ) represents γ1 (s) ≤ γ2 (s) (or γ1 (s) < γ2 (s)) for all s > 0. Id represents identity function. We employ some notations in graph theory [28] to describe the interconnection of the dynamical network. The interconnection of the dynamical network can be represented with a directed graph by considering the subsystems as vertices. We use Vi to represent the i-th vertex (subsystem) in the graph. If the state of the i-th subsystem is an input of the j-thsubsystem, then γji ̸= 0 and  there is a directed arc (interaction) Vi Vj from Vi to Vj ; otherwise, γji = 0 and there is no arc from Vi to Vj . A path in the directed graph is any sequence of arcs where the final  vertex of one is the initial vertex of the next one, denoted by Vi Vj · · · Vk . If there exists a path leading from Vi to Vj , then Vj is reachable from  Vi . Specifically, Vj is reachable from itself. The reaching set RS Vj is the set of the vertices from which Vj is reachable. A closed path with no repeated vertices other than the starting and ending vertices is called a simple cycle. 3. System formulation

xi (T + 1) = fi (x (T ) , ui (T )) ,

i = 1, . . . , N ;

(1) ∑N

where xi ∈ R , x = [ , . . . , ] ui ∈ R and fi : R → Rni is continuous. T takes values in Z+ . Assume that fi (0, 0) = 0, and the external input u = [uT1 , . . . , uTN ]T is bounded. Denote f (x, u) = [f1T (x, u1 ), . . . , fNT (x, uN )]T . It is assumed that for each xi -subsystem (i = 1, . . . , N), there exists a continuous ISS-Lyapunov function Vi such that the following properties hold. xT1

(2) there exist αi ∈ K∞ , σij ∈ K∞ ∪ {0} and σui ∈ K ∪ {0} such that Vi (fi (x (T ) , ui (T ))) − Vi (xi (T ))

≤ −αi (Vi (xi (T )))      + max σij Vj xj (T ) , σui (|ui (T )|) .

(3)

j=1,...,N ;j̸=i

Without loss of generality, we assume that (Id − αi ) ∈ K as in [9]. Define

γˆij = αi−1 ◦ (Id − ρi )−1 ◦ σij

(4)

as the ISS gain from Vj to Vi , with ρi positive definite satisfying (Id − ρi ) ∈ K∞ . Correspondingly, we define

γˆui = αi−1 ◦ (Id − ρi )−1 ◦ σui .

(5)

Remark 1. There are two Lyapunov formulations of discrete-time ISS systems: ‘‘gain margin’’ form and dissipation form (refer to [9,29]). The dissipation form is in the form of (3), while the ‘‘gain margin’’ form can be described as there exist a continuous and positive definite αi′ and γij′ , γui′ ∈ K ∪ {0} such that Vi (xi (T )) ≥ max γij′ Vj xj (T )

 , γui′ (|ui (T )|) ⇒ Vi (fi (x (T ) , ui (T ))) − Vi (xi (T )) 

 



≤ −αi′ (Vi (xi (T ))) .

(6)

However, different from continuous-time systems, the trajectory of the discrete-time system may ‘‘jump’’ out of the region determined by the ‘‘gain margin’’, which means that γij′ and γui′ may not be the real nonlinear gains from Vj ’s and ui to Vi . However, as done in [12,13], to prove small-gain results the ‘‘gain margin’’ form formulation (6) should be consolidated with Vi (xi (T )) ≤ max γij′ Vj xj (T )



 



 , γui′ (|ui (T )|)

⇒ Vi (fi (x (T ) , ui (T )))       ≤ Id − δi′ max γij′ Vj xj (T ) ,  γui′ (|ui (T )|)

(7)

where δi′ is continuous and positive definite satisfying Id − δi ∈ K∞ . 



 ′

Combining (6) and (7), the ‘‘gain margin’’ formulation for smallgain analysis is obtained by replacing property (2) with

The discrete-time dynamical network studied in this paper is composed of N subsystems in the following form

ni

267

xTN T ,

nui

j=1 nj +nui

(1) there exist α i , α i ∈ K∞ such that

α i (|xi |) ≤ Vi (xi ) ≤ α i (|xi |) ,

∀xi ;

(2)

(2’) there exist γij ∈ K∞ ∪ {0} and γui ∈ K∞ ∪ {0} such that Vi (fi (x (T ) , ui (T )))

     γij Vj xj (T ) ,  ≤ (Id − δi )  max Vi (xi (T )) , j∈{1,...,N }\{i}  γui (|ui (T )|)  

(8)

where δi is continuous and positive definite satisfying (Id − δi ) ∈ K∞ . For the sake of generality of our main result, we will first consider the dissipation formulation, and then extend the result to the ‘‘gain margin’’ form following a similar idea.

268

T. Liu et al. / Systems & Control Letters 61 (2012) 266–272

4. Systems formulated in dissipation form 4.1. Main result We employ the idea of ‘‘potential influence’’ in [23] to construct a Lyapunov function candidate for discrete-time dynamical network (1). In the dynamical network, the potential influence acting on the p-th subsystem from the other subsystems can be described as

V (x) = [p]



[p] Vj

(x)

(9)

j=1,...,N

with [p] Vj





(x) = γˆ0i[p] ◦ γˆi[p] i[p] ◦ · · · ◦ γˆi[p] 1

[p]

where i1

1

Vi[p]

[p] i j−1 j

2

[p]

= p, γˆ0i[p] = Id, ik

 j

 xi[p] j

∈ {1, . . . , N }, k ∈ {1, . . . , j},

1

[p] [p] ik ̸= ik′ if k ̸= k′ , for j = 1, . . . , N. Here we introduce the γˆ0i[p] for 1

convenience of notation. Define the following ISS-Lyapunov function candidate for discrete-time dynamical network (1):

 VΠ (x) = max VΠ (x) = max

 

V

[p]

(x)

(10)

p∈Π

where the set Π ⊆ {1, . . . , N } satisfies = p∈Π RS Vp {V1 , . . . , VN }. It can be easily verified that VΠ is positive definite and radially unbounded with respect to x. Correspondingly, the potential influence of the external input



u = uT1 , . . . , uTN



T



U[p] (T ) =







acting on the p-th subsystem is described as [p]

Uj (T )

(11)

j=1,...,N

with [p]



Uj (T ) = γˆ0i[p] ◦ γˆi[p] i[p] ◦ · · · ◦ γˆi[p] 1

1

[p] i j−1 j

2

    ◦ γˆui[p] ui[p] (T ) j j

Fig. 1. The j subsystems on a specified path ma ending at Via . 1

[24,25] for general nonlinear systems. Define

γ11 (r1 ) · · ·  .. Γ (r ) = MAX  ... . γN1 (r1 ) · · ·   max {γ1i (ri )} i=1,...,N   ..  =  .   max {γNi (ri )}

Q (r ) = MAX r , Γ (r ), . . . , Γ (N −1) (r )



u¯ Π (T ) = max UΠ = max

U (T ) .

(15)

with r = [r1 , . . . , rN ]T for r1 , . . . , rN ∈ R+ . If the dynamical network (1) satisfies the cyclic-small-gain condition (13), then we can construct a vector Lyapunov function as Vvec (x) = Q ([V1 (x1 ), . . . , VN (xN )]T ). Then, we can construct a single ISSLyapunov function as Vs (x) = maxVvec (x). Considering the definition of VΠ in (10), direct calculation yields Vs (x) = VΠ (x) with Π = {1, . . . , N }. Although the construction of Vvec and thus Vs could be simplified given the gain interconnection structure, the construction of VΠ in (10) takes the probability of Π ⊂ {1, . . . , N } into full consideration, which might be more easily computable and more flexible for dynamical networks with specific gain interconnection structures. Without affecting the validity of the main result, we will prove Theorem 1 by considering u¯ Π as the new input of the dynamical network.

Viaj · · · Via2 Via1

(as shown in Fig. 1). From the

definitions of VΠ (x) and u¯ Π , we have (12)

p∈Π

It can be easily verified that u¯ Π (T ) is positive definite and radially unbounded with respect to [γu1 (|u1 (T )|) , . . . , γuN (|uN (T )|)]T for all u. Our first main result is summarized in Theorem 1. Theorem 1. Consider the discrete-time dynamical network composed of N subsystems in the form of (1). Suppose each i-th subsystem (i = 1, . . . , N) admits a continuous ISS-Lyapunov function Vi satisfying properties (2) and (3). Then, the dynamical network is ISS with u = [uT1 , . . . , uTN ]T as the input and VΠ defined in (10) as an ISS-Lyapunov function, if there exist positive definite functions ρi satisfying (Id − ρi ) ∈ K∞ such that for each r = 2, . . . , N, the ISS gain functions γˆij defined in (4) satisfy

γˆi1 i2 ◦ γˆi2 i3 ◦ · · · ◦ γˆir i1 < Id



Consider any arbitrary element   in VΠ (x(T )), which corresponds

 [p]

(14)

i=1,...,N

to a path M a =



γNN (rN )

4.2. Proof of Theorem 1

for j = 1, . . . , N. Define



 γ1N (rN ) ..  . 



(13)

for all 1 ≤ ij ≤ N, ij ̸= ij′ if j ̸= j′ . Remark 2. It should be noted that we can also construct an ISSLyapunov function for the discrete-time dynamical network (1) based on the vector Lyapunov function approach proposed in

   γˆ0ia1 ◦ γˆia1 ia2 ◦ · · · ◦ γˆiak−1 iak ◦ · · · ◦ γˆiaj−1 iaj Viaj xiaj (T ) ≤ VΠ (x (T ))

(16)

and

    γˆ0ia1 ◦ γˆia1 ia2 ◦ · · · ◦ γˆiaj−1 iaj ◦ γˆuiaj uiaj (T ) ≤ u¯ Π (T ) .

(17)

If j = 1, then the path M a contains only Via . If j ≥ 2, then for all 1 k ∈ {1, . . . , j − 1}, we have

   γˆ0ia1 ◦ γˆia1 ia2 ◦ · · · ◦ γˆiak−1 iak Viak xiak (T ) ≤ VΠ (x (T )) ,

(18)

i.e.,





1 1 Via xia (T ) ≤ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 (VΠ (x (T ))) . a a a ◦ γ ia i k

k

k−1 k

1 2

(19)

1

Using the cyclic-small-gain condition, we have

γˆiak iak+1 ◦ · · · ◦ γˆiaj−1 iaj ◦ γˆiaj iak < Id,

(20)

i.e., 1 1 γˆiaj iak < γˆi− ◦ · · · ◦ γˆi− a a a ia i j−1 j

for k = 1, . . . , j − 1.

k k+1

(21)

T. Liu et al. / Systems & Control Letters 61 (2012) 266–272

269





1 1 ◦ · · · ◦ γˆi− ˆ0i−a1 + Id − ρiaj ◦ αiaj ◦ γˆi− a a ◦ γ a i ia

Combining (19) and (21), we have

j−1 j





γˆiaj iak Viak xiak (T )



1 1 ˆ0i−a1 (VΠ (x (T ))) ◦ · · · ◦ γˆi− ≤ γˆiaj iak ◦ γˆi− a a ◦ γ a i ia k−1 k

j−1 j

1

1 2

k−1 k

k k+1

1

1

1 2

− ρiaj ◦ αiaj ◦ γˆia

−1

1 1 1 1 ˆ0i−a1 (VΠ (x (T ))) ◦ · · · ◦ γˆi− ◦ γˆi− ◦ · · · ◦ γˆi− < γˆi− a a ◦ γ a a a a i ia i ia j−1 j

1 2

× (VΠ (x (T ))) 1 1 ˆ0i−a1 (VΠ (x (T ))) ◦ · · · ◦ γˆi− = γˆi− a a ◦ γ a i ia ia j−1 j

1 ◦ · · · ◦ γˆi− ˆ0i−a1 (VΠ (x (T ))) . a a ◦ γ i 1 2

1

1

1 2

(31)

(22)

With Lemma A.2 of Appendix, by considering Via (xia (T +

i.e.,

1 1 ˆ0i−a1 (VΠ (x (T ))) as s′ , ρia ◦ αia ◦ · · · ◦ γˆi− 1)) as s, γˆi− a a ◦ γ a i ia



j−1 j

j−1 j

1

1 2

1 2

definite function α˜ M a such that

1 1 ˆ0i−a1 (VΠ (x (T ))) ◦ · · · ◦ γˆi− ≤ −α˜ M a ◦ γˆi− a a ◦ γ a i ia j−1 j

(24)



γˆ0ia1 ◦ γˆia1 ia2 ◦ · · · ◦ γˆiaj−1 iaj ◦ γˆiaj l (Vl (xl (T ))) ≤ VΠ (x (T ))

(25)

(32)

i.e.,

   γˆ0ia1 ◦ γˆia1 ia2 ◦ · · · ◦ γˆiaj−1 iaj Viaj xiaj (T + 1) ≤ (Id − α¯ M a ) (VΠ (x(T )))

(33)

where α¯ M a is positive definite and satisfies (Id − α¯ M a ) ∈ K∞ . Case 2: VΠ (x (T )) < u¯ Π (T ). Property (17) can be rewritten as

i.e.,

γˆ (Vl (xl (T ))) ≤ γˆia

−1

iaj l

ia j−1 j

◦ · · · ◦ γˆia ia ◦ γˆ0ia (VΠ (x (T ))) . −1

−1

1 2

1

(26)

Equivalently,

σiaj l (Vl (xl (T )))   1 1 ≤ Id − ρiaj ◦ αiaj ◦ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 (VΠ (x (T ))) a a a ◦ γ ia i j−1 j

1 2

Id − ρia

    σuiaj uiaj (T )   1 1 ≤ Id − ρiaj ◦ αiaj ◦ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 a a a ◦ γ ia i j−1 j

(27)

1

1

(34)

From property (28), one can observe



 a

Id − ρia



1 1 ◦ αiaj ◦ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 (¯uΠ (T )) a a a ◦ γ ia i j−1 j 1 2 1   max   σia l (Vl (xl (T ))) . j j



1 1 ◦ αiaj ◦ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 (VΠ (x (T ))) a a a ◦ γ ia i j−1 j 1 2 1   max   σia l (Vl (xl (T ))) . j

≥

(35)

l∈{1,...,N }\ iaj

j

(28)

l∈{1,...,N }\ iaj

Combining (3), (16), (34) and (35), we obtain





Via xia (T + 1) j

Case 1: VΠ (x (T )) ≥ u¯ Π (T ). Using (17), we have j−1 j

1 2

j



    1 1 γˆuiaj uiaj (T ) ≤ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 (VΠ (x (T ))) (29) a a a ◦ γ ia i 1

≤ Id − αiaj





Via xia (T ) j



j



 

 

max   σia l (Vl (xl (T ))) , σuia uia (T ) j j j

+

l∈{1,...,N }\ iaj

i.e.,

    σuiaj uiaj (T )   1 1 ≤ Id − ρiaj ◦ αiaj ◦ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 a a a ◦ γ ia i j−1 j

1 2



 1 1 ≤ Id − αiaj ◦ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 (¯uΠ (T )) a a a ◦ γ ia i j−1 j 1 2 1   1 − 1 ˆ0i−a1 (¯uΠ (T )) + Id − ρiaj ◦ αiaj ◦ γˆia ia ◦ · · · ◦ γˆi− a a ◦ γ i

1

× (VΠ (x (T ))) .

j−1 j

(30)



Via xia (T + 1)



≤ Id − α

iaj

+

1 2

V

 iaj

max  

l∈{1,...,N }\ iaj



x (T )



iaj

    σiaj l (Vl (xl (T ))) , σuiaj uiaj (T )

  1 1 ≤ Id − αiaj ◦ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 (VΠ (x (T ))) a a a ◦ γ ia i j−1 j

1

1 1 − ρiaj ◦ αiaj ◦ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 a a a ◦ γ ia i j−1 j



1

1 2

1

× (¯uΠ (T )) .

j



1 2

1 1 ◦ · · · ◦ γˆi− ˆ0i−a1 (¯uΠ (T )) = γˆi− a a a ◦ γ ia i j−1 j

Combining (3), (16), (28) and (30), we have j

1 2

× (¯uΠ (T )) .

holds for all l ∈ {1, . . . , N } \ i1 , ia2 , . . . , ij . Properties (24) and (27) together imply that

a

≥

1

1 2

≤ −α¯ M a (VΠ (x(T ))) ,

1

for all l ∈ ia1 , ia2 , . . . , iaj−1 . If j = N, then the path M a contains all the subsystems in the dynamical network. If j ≤ N − 1, then for all l ∈ {1, . . . , N } \  a a i1 , i2 , . . . , iaj , we have



as χ , there exists a positive

   γˆ0ia1 ◦ γˆia1 ia2 ◦ · · · ◦ γˆiaj−1 iaj Viaj xiaj (T + 1) − VΠ (x(T ))

σiaj l (Vl (xl (T )))   1 1 ◦ · · · ◦ γˆi− ˆ0i−a1 (VΠ (x (T ))) < Id − ρiaj ◦ αiaj ◦ γˆi− a a ◦ γ a i ia j−1 j

1 2

1

(23)

ia j−1 j

j

j

1

1 2

as α and γˆ0ia ◦ γˆia ia ◦ · · · ◦ γˆia

for all k = 1, . . . , j − 1. Equivalently,



j

j

 σiaj iak Viak xiak (T )   1 1 ˆ0i−a1 (VΠ (x (T ))) ◦ · · · ◦ γˆi− < Id − ρiaj ◦ αiaj ◦ γˆi− a a ◦ γ a i ia 

1 2

1

(36)

With Lemma A.2 of Appendix, similarly as in Case 1, one can achieve

   γˆ0ia1 ◦ γˆia1 ia2 ◦ · · · ◦ γˆiaj−1 iaj Viaj xiaj (T + 1) − u¯ Π (T ) ≤ −α¯ M a (¯uΠ (T )) ,

(37)

270

T. Liu et al. / Systems & Control Letters 61 (2012) 266–272

Using definitions of VΠ (x) and u¯ Π , we have

i.e.,

γˆ0ia1 ◦ γˆia1 ia2 ◦ · · · ◦ γˆiaj−1 iaj



 Via (x(T + 1)) j

(38) ≤ (Id − α¯ M a ) (¯uΠ (T )) .    Note that γˆ0ia ◦γˆia ia ◦· · ·◦γˆia ia Via xia (T + 1) is an arbitrary j j j−1 j 1 2 1 element in VΠ (x (T + 1)). Considering both Cases 1 and 2, by choosing α¯ Π as the minimum of the functions α¯ M a corresponding to all the paths (elements) in VΠ (x (T + 1)), we have VΠ (x (T + 1)) ≤ (Id − α¯ Π ) (max {VΠ (x (T )) , u¯ Π (T )})





1 1 ◦ · · · ◦ γˆi− ˆ0i−a1 (VΠ (x (T ))) Via xia (T ) ≤ γˆi− a a ◦ γ a i ia

(42)

    1 1 ◦ · · · ◦ γˆi− ˆ0i−a1 (¯uΠ (T )) . γˆuiaj uiaj (T ) ≤ γˆi− a a ◦ γ a i ia

(43)

j

j

j−1 j

1 2

j−1 j

1

1 2

1

With (41)–(43) satisfied, property (8) in Section 3 implies





Via xia (T + 1) j

j

 1 1 ˆ0i−a1 ◦ · · · ◦ γˆi− ≤ Id − δiaj ◦ γˆi− a a ◦ γ a i ia 

(39)

j−1 j

where α¯ Π is positive definite and satisfies (Id − α¯ Π ) ∈ K∞ . The ISS of the dynamical network follows the ‘‘gain margin’’ Lyapunov formulation of ISS as discussed in Remark 1.

1

1 2

× (max {VΠ (x (T )) , u¯ Π (T )}) .

(44)

1 1 ◦ · · · ◦ γˆi− With Lemma A.3 of Appendix, by considering γˆi− a a ◦ a i ia j−1 j

Remark 3. Denote the largest element of VΠ at time T as VΠ∗ (x∗ (T )). It has been shown in [12] that the behavior of a discrete-time feedback system is determined by all the subsystems, i.e., VΠ (x (T + 1)) is determined by all the elements of VΠ but not just by VΠ∗ (x∗ (T + 1)). Roughly speaking, for discrete-time dynamical networks, we proved that for any elements VΠa ∈ VΠ , VΠa (xa (T + 1)) < VΠ∗ (x∗ (T )). This is one of the major differences between continuous-time interconnected systems and discretetime interconnected systems.  Remark 4. In the proof of Theorem 1, VΠ is shown satisfying the ‘‘gain margin’’ formulation of ISS. A dissipation form ISS-Lyapunov function can be further constructed based on VΠ with similar ideas as in Remark 3.3 in [9], and Proof of Proposition 2.6 in [30]. Notice that αΠ in (39) should be of class K∞ to apply these methods. This problem can be solved with Lemma 2.8 in [31]. 





Via xia (T + 1) j

j

  1 −1 −1 ¯ ≤ γˆi− ˆ ˆ a a ◦ ··· ◦ γ a a ◦ γ a ◦ Id − δia i i i 0i j j−1 j

1 2

1

× (max {VΠ (x (T )) , u¯ Π (T )})

(45)

i.e.,

   γˆ0ia1 ◦ · · · ◦ γˆiaj−1 iaj Viaj xiaj (T + 1)   ≤ Id − δ¯iaj (max {VΠ (x (T )) , u¯ Π (T )}) .

(46)

Define δ¯ (s) = mini∈{1,...,N } δ¯ i (s) for s ≥ 0. It is clear that δ¯





is continuous and positive definite, and satisfies Id − δ¯ ∈ K∞ . Noting that M a corresponds to any arbitrary element in VΠ (x), we can achieve



5. Systems formulated in ‘‘gain margin’’ form In this section, we provide a counterpart of the result in Section 4 for dynamical networks with subsystems formulated in ‘‘gain margin’’ form.

1 2

γˆ0i−a1 as χ and δiaj as ε , there exists a continuous, positive definite δ¯iaj 1   satisfying Id − δ¯ ia ∈ K∞ such that j



VΠ (x(T + 1)) ≤ Id − δ¯ (max {VΠ (x (T )) , u¯ Π (T )}) .





(47)

Theorem 2 is proved. 5.1. Main result Let γˆ(·) = γ(·) . We still construct an ISS-Lyapunov function candidate in the form of (10) for dynamical networks with subsystems formulated in ‘‘gain margin’’ form. Theorem 2. Consider the discrete-time dynamical network composed of N subsystems in the form of (1). Suppose each i-th subsystem (i = 1, . . . , N) admits a continuous ISS-Lyapunov function Vi satisfying properties (2) and (8). Then, the dynamical network is ISS with u = [uT1 , . . . , uTN ]T as the input and VΠ defined in (10) as an ISSLyapunov function candidate if for each r = 2, . . . , N

γi1 i2 ◦ γi2 i3 ◦ · · · ◦ γir i1 < Id

(40)

for all 1 ≤ ij ≤ N, ij ̸= ij′ if j ̸= j′ . 5.2. Proof of Theorem 2 Similarly with the proof of Theorem 1, consider any arbitrary element in VΠ (x(T )), which corresponds to a path M a = (Via j

· · · Via2 Via1 ). With similar approaches as for properties (22) and (26), we can ultimately obtain

1 1 γˆiaj l (Vl (xl (T ))) ≤ γˆi− ◦ · · · ◦ γˆi− ˆ0i−a1 (VΠ (x (T ))) a a a ◦ γ ia i j−1 j 1 2 1  a for all l ∈ {1, . . . , N } \ ij .

6. An example We employ a numerical example to show how to construct an ISS-Lyapunov function for a discrete-time dynamical network. Consider the following dynamical network composed of three subsystems: x1 (T + 1) = 0.6x1 (T ) + max{0.36x32 (T ), 3.2x33 (T ), u1 (T )}

(48)



 1 x2 (T + 1) = 0.4x2 (T ) + max 0.6x13 (T ), 1.2x3 (T ), u2 (T ) 

1

(49)



x3 (T + 1) = 0.2x3 (T ) + max 0.36x13 (T ), 0.36x2 (T ), u3 (T )

(50)

where x1 , x2 , x3 ∈ R are the states of the subsystems. Denote x = [x1 , x2 , x3 ]T and u = [u1 , u2 , u3 ]T . Define the following ISS-Lyapunov function candidate for each subsystem: Vi (xi ) = |xi |.

(51)

Then, direct calculation yields (41)

Vi (xi (T + 1)) − Vi (xi (T ))

= −αi (Vi (xi (T ))) +

max

{σij (Vj (xj (T ))), σui (ui (T ))}

j∈{1,2,3}\{i}

(52)

T. Liu et al. / Systems & Control Letters 61 (2012) 266–272

Fig. 2. State trajectories of the dynamical network.

271

Fig. 3. The evolutions of VΠ and uΠ on discrete-time.

7. Conclusions

where

α1 (s) = 0.4s,

σ12 (s) = 0.36s3 ,

σ13 (s) = 3.2s3 ,

σu1 (s) = s,

α2 (s) = 0.6s, σ23 (s) = 1.2s,

σ21 (s) = 0.6s 3 , σu2 (s) = s,

α3 (s) = 0.8s, σ32 (s) = 0.36s,

σ31 (s) = 0.36s 3 , σu1 (s) = s,

1

(53)

1

for s ∈ R+ . Choose ρ1 (s) = ρ2 (s) = ρ3 (s) = 0.02s for s ∈ R+ . Then, it is easy to obtain the gain functions for the subsystems:

γˆ12 (s) = 0.9184s3 ,

γˆ13 (s) = 8.1633s3 ,

1 3

γˆ23 (s) = 2.0408s,

1 3

γˆ32 (s) = 0.4592s

γˆ21 (s) = 1.0204s , γˆ31 (s) = 0.4592s ,

(54)

This paper presents a Lyapunov formulation of the ISS cyclicsmall-gain criterion for discrete-time dynamical networks. It can be shown that the cyclic-small-gain theorem can recover both the conventional small-gain results and the recently developed smallgain results for dynamical networks. Due to wide applications of discrete-time system theory in control engineering, it is expected that the Lyapunov-based ISS cyclic-small-gain results presented in this paper will prove useful for the analysis and design of complex dynamic networks. Acknowledgments We would like to thank the associate editor and anonymous reviewers for constructive comments which led to the improvement of the paper. Appendix. Technical Lemmas

for s ∈ R+ . In addition, these gain functions satisfy

γˆ12 ◦ γˆ21 < Id,

γˆ23 ◦ γˆ32 < Id,

γˆ12 ◦ γˆ23 ◦ γˆ31 < Id,

γˆ31 ◦ γˆ13 < Id,

γˆ13 ◦ γˆ32 ◦ γˆ21 < Id.

(55)

Therefore, the discrete-time dynamical network (48)–(50) satisfies the cyclic-small-gain condition. By choosing Π = {1}, we construct the following ISS-Lyapunov function for the dynamical network: VΠ (x) = max{V1 (x1 ), γˆ12 (V2 (x2 )), γˆ13 ◦ γˆ32 (V2 (x2 )),

γˆ13 (V3 (x3 )), γˆ12 ◦ γˆ23 (V3 (x3 ))} = max{V1 (x1 ), 0.9184V23 (x2 ), 8.1633V33 (x3 )}.

(56)

Correspondingly, uΠ = max{σu1 (u1 ), γˆ12 ◦ σu2 (u2 ), γˆ13 ◦ γˆ32 ◦ σu2 (u2 ),

γˆ13 ◦ σu3 (u3 ), γˆ12 ◦ γˆ23 ◦ σu3 (u3 )} = max{2.5|u1 |, 4.2516|u2 |3 , 15.9439|u3 |3 }.

(57)

Figs. 2 and 3 show the simulation results with initial condition x(0) = [0.6, 0.8677, 0.4189]T and inputs u(T ) = [0.1 sin(5T ), 0.1 sin(6T ), 0.1 sin(7T )]T . The evolutions of VΠ and uΠ shown in Fig. 3 are in accordance with the theoretical result that VΠ is an ISS-Lyapunov function of the dynamical network.

Lemma A.1. For χi1 , χi2 ∈ K ∪ {0} (i = 1, . . . , n with n a positive integer), if χi1 ◦ χi2 < Id for i = 1, . . . , n, then there exists a positive definite function η such that (Id − η) ∈ K∞ and χi1 ◦ (Id − η)−1 ◦ χi2 < Id for i = 1, . . . , n. Proof. Recall the fact that for any χ1 , χ2 ∈ K ∪{0}, χ1 ◦χ2 < Id ⇔ χ2 ◦ χ1 < Id. Property χi1 ◦ (Id − η)−1 ◦ χi2 < Id is equivalent to (Id − η)−1 ◦ χi2 ◦ χi1 < Id. Define χ0 (s) = min{ 12 (χi1−1 ◦ χi2−1 (s) + s)} for s ≥ 0. Obviously, χ0 ∈ K∞ . For all i = 1, . . . , n, because χi2 ◦ χi1 < Id, we have χi1−1 ◦ χi2−1 > Id. Thus, χ0 > Id. We also have χ0 ◦ χi2 ◦ χi1 ≤ 1 (Id + χi2 ◦ χi1 ) < Id for all i = 1, . . . , n. Define η¯ = χ0 − Id. 2 Then, η¯ is positive definite, (Id + η) ¯ ∈ K∞ , and (Id + η) ¯ ◦ χi2 ◦ χi1 < Id for i = 1, . . . , n. The proof follows readily by defining η = Id − (Id + η) ¯ −1 , or equivalently η = η¯ ◦ (Id + η) ¯ −1 .  Lemma A.2. For any positive definite function α , and for any class K∞ function χ , there a positive definite function α˜ such that  exists  χ s′ − χ (s) ≥ α˜ s′ for any pair of nonnegative numbers s, s′ satisfying s′ − s ≥ α s′ .

 

Proof. s′ − s ≥ α s′ can be written as (Id − α) s′ ≥ s. Assume one can find a smaller α ′ to replace α (Id − α) ∈ K . (Otherwise,  ′ such that Id − α ∈ K .)

 

 

272

T. Liu et al. / Systems & Control Letters 61 (2012) 266–272

Note that χ −1 ◦χ ◦(Id − α) = Id −α < Id implies χ ◦(Id − α)◦ χ −1 < Id. With Lemma A.1, we can find a positive definite function α¯ satisfying (Id − α) ¯ ∈ K∞ , such that

¯ −1 ◦ χ ◦ (Id − α) ◦ χ −1 < Id. (Id − α)

(58)

Consequently,

χ ◦ (Id − α) < (Id − α) ¯ ◦ χ.

(59)

Define α0 = α¯ ◦ χ . Then, α0 is positive definite and for any one positive definite function α ′ ≤ α0

     ¯ ◦ χ s′ χ − α ′ s′ = (Id − α)   ≥ χ ◦ (Id − α) s′ ≥ χ (s) (60)  ′ holds for all pair of nonnegative numbers s, s satisfying s′ − s ≥   α s′ .  Lemma A.3. For any class K∞ function χ and any continuous, positive definite function ε satisfying (Id − ε) ∈ K∞ , there exists a continuous, positive definite function µ satisfying (Id − µ) ∈ K∞ such that χ ◦ (Id − µ) = (Id − ε) ◦ χ . Proof. (Id − ε)◦χ ◦χ −1 = Id −ε < Id implies χ −1 ◦(Id − ε)◦χ < Id. The result is proved by defining µ = Id −χ −1 ◦(Id − ε)◦χ .  References [1] D.J. Hill, A generalization of the small-gain theorem for nonlinear feedback systems, Automatica 27 (6) (1991) 1043–1045. [2] I.M.Y. Mareels, D.J. Hill, Monotone stability of nonlinear feedback systems, Journal of of Mathematical Systems, Estimation, and Control 2 (3) (1992) 275–291. [3] Z.P. Jiang, A.R. Teel, L. Praly, Small-gain theorem for ISS systems and applications, Mathematics of Control, Signals and Systems 7 (1994) 95–120. [4] E.D. Sontag, Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control 34 (1989) 435–443. [5] E.D. Sontag, Further facts about input to state stabilization, IEEE Transactions on Automatic Control 35 (1990) 473–476. [6] E.D. Sontag, Y. Wang, On characterizations of the input-to-state stability property, System & Control Letters 24 (1995) 351–359. [7] A.R. Teel, A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE Transactions on Automatic Control 41 (9) (1996) 1256–1270. [8] Z.P. Jiang, I.M.Y. Mareels, A small-gain control method for nonlinear cascade systems with dynamic uncertainties, IEEE Transactions on Automatic Control 42 (3) (1997) 292–308. [9] Z.P. Jiang, Y. Wang, Input-to-state stability for discrete-time nonlinear systems, Automatica 37 (2001) 857–869. [10] Z.P. Jiang, Y. Lin, Y. Wang, Nonlinear small-gain theorems for discrete-time feedback systems and applications, Automatica 40 (2004) 2129–2136.

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