A Unified Stability Analysis Approach for a Class of Interconnected System
aa r X i v : . [ c s . S Y ] M a y A Unified Stability Analysis Approach for A Class ofInterconnected System
WANG Yong
Beijing Institute of Control Engineering, Beijing 100190, P. R. ChinaE-mail: [email protected]
Abstract:
From the structural perspective, this paper investigates a new formulation of the concept of input-to-state stability (ISS),and based on this formulation, proposes a new stability analysis approach for a class of interconnected system. The new formulationof ISS is better able to reflect the tendency of the state x ( t ) tracking the input u ( t ) and weakens the conservative of the original form.The stability analysis method which transforms the interconnected system into the equivalent cascade form, does not depend on theLyapunov function, breaks through the limitation of the small-gain theorem and extends the application of ISS. As its applications inthree typical kinds of interconnected systems, this method is used to prove the small-gain theorem again and analyzes the stability ofa class of interconnected system and the consensus of the multi-agent system (MAS). Key Words: input-to-state stability, interconnected system, cascade system, small-gain theorem, stability analysis, multi-agentsystem..
The concept of the input-to-state stability (ISS) was intro-duced by E.D.Sontag in 1989 in the well-known paper[1],which becomes a popular method to study the input-outputproperty of nonlinear systems later. Generally, a system ˙ x = f ( t, x, u ) is said to be ISS if there exist class KL function β and class K function γ , such that for any initial value inthe closed set D and the bounded input u ( t ) , the following issatisfied: k x ( t ) k ≤ β ( k x k , t − t ) + γ ( sup τ ∈ [ t ,t ) k u ( τ ) k ) ISS describes the evolution of the state of a stable systemwhen it is driven by the external input. Based on the con-cept, many researchers proposed some other concepts, suchas IOS(input to output stable),OLIOS(output-Lagrange inputto output stable),SIIOS(state-independent IOS), ROS(robustlyoutput stable)[2],iISS (integral input-to-state stable)[3] , andISDS (input-to-state dynamically stable) [4] which describesthe dynamic process of a stable system. The introduction ofISS gives us a new way to study the stability of a system. Formore complicated systems, which contain many subsystemsthat interconnect with each other, if every subsystem is ISS,we can take a structural perspective, ignore their internal de-tails and take full advantage of the input-output property andinterconnected relationship between subsystems to study thestability problem. In this way, we only need to focus on the re-lationship between all subsystems and need not care about alldetails,which is greatly different from the Lyapunov functionbased method. Therefore, the concept of ISS greatly simplifiesthe stability analysis of the complex interconnected system.
This research is supported by the National Natural Science Foundation ofChina under grant 61333008 and the National Basic Research Program (973)of China under Grant 2013CB733100.
A good example of the above idea is the well-known small-gain theorem. In short, if two subsystems are ISS and inter-connected with each other, if the composition of the gain func-tion (the quantitative expression of the input-output property ofeach subsystem)along the closed cycle is less than the identityfunction, the entire system is stable. Later on, with ISS as tool,many researchers extended the small-gain theorem from linearsystems to nonlinear systems and proposed its various formsand the associated proofs [5]-[7]. Furthermore, Jiang and etc.extend the small-gain theorem to the case of the interconnectedsystem with more than two subsystems, and give the sufficientcondition that ensures stability in [6], i.e., if the composition ofthe gain function along every closed cycle is less than the iden-tity function, the entire system is stable. Because the structuralperspective is brief and intuitionistic, it becomes an importantmethod to design the controller [8][9]and analyze the stability. However, when facing some new problems, the above con-cept and approach meet across some difficulties .1) The form of ISS is too conservative to describe the corre-sponding change of the state x ( t ) of system tracking the input u ( t ) . sup τ ∈ [ t ,t ) k u ( τ ) k just denotes the maximum of u ( t ) in a certaintime interval, but in fact, x ( t ) keeps tracking the change of theinput u ( t ) all the time so as to be kept in a neighboring area ofit. Therefore sup τ ∈ [ t ,t ) k u ( τ ) k is too conservative to describe thefact, especially, when u ( t ) converges towards a constant.2) Some systems, only one of whose subsystems is ISS, arestill stable in fact.The small-gain theorem requires all subsystems should beISS, but in fact, some systems like the one given by equation(1) below, are also stable even though the first subsystem is notISS. xample 1. (cid:26) ˙ x = z ˙ z = − z − x (1)where x, z ∈ R . The z-subsystem,i.e. the second equation, isISS, but the x-subsystem,i.e. the first equation, is not.3) Even though every agent is ISS, the consensus prob-lem of multi-agent systems is not explained via the small-gaintheorem.Since the composition of the gain function along theclosed cycle equals the identity function,which conflicts withthe small-gain theorem.Therefore, from the structural perspective, the tools in thecurrent literature can not solve these new problems, which ur-gently requires to develop a new formulation of ISS and a newapproach. In this paper, we investigate a new formulation of ISS, andbased on it, propose a stability analysis approach for a class ofinterconnected system. The contribution of this paper includesthree parts. Firstly, a new formulation of ISS is proposed,which is better able to reflect the corresponding change of x ( t ) tracking u ( t ) than the original form, and using this new formu-lation, we uncover the essential relationship between the inter-connected system and the cascade system. Secondly, basedon the research of ISS, a unified stability analysis approach isproposed, which does not depend on the construction of a Lya-punov function. Thirdly, as the application of this approach,we analyze the stability of three typical kinds of interconnectedsystems, i.e., repeating to prove the small-gain theorem in thisnew framework and analyzing the stability of a class of in-terconnected system like equation (1) and the consensus of amulti-agent system. The rest of paper is organized as follows. Section 2 givessome notations and briefly recalls some basic backgroundknowledge. Section 3 presents the framework of this approach.Section 4 presents a new formulation of ISS. Section 5 gives itsapplications for three typical interconnected systems. Section6 is the conclusion and talks about other applications.
Classes of
K, K ∞ , KL and positive definite function fol-low the definition in [10], which are extensively used in thefield. D ∈ R n denotes a domain containing the origin. Now, we recall the traditional concept of ISS again. Consider thegeneral nonlinear system ˙ x = f ( x, u ) (2)where f : R n × R m → R n is the continuous function andlocal Lipschitz function w.r.t. x ( t ) and u ( t ) .The Lyapunov-like theorem that follows gives a sufficientand necessary condition for ISS. Proposition 1. [10] Let V : [0 , ∞ ] × R n → R be a continu-ously differentiable function such that α ( k x k ) ≤ V ≤ α ( k x k ) (3) ˙ V ≤ − W ( x ) , ∀ k x k ≥ ρ ( k u k ) > (4)where α , α ∈ K ∞ , ρ ∈ K ,and W is continuous positivedefinite function on R n .Then ,the system (2) is input-to-statestable with γ = α − ◦ α ◦ ρ . In this paper, we investigate the stability problem of a classof interconnected system described by the followingSubsystem x : ˙ x = f ( x, z ) , (5)Subsystem z : ˙ z = f ( z, x ) , (6)where x ∈ R n , z ∈ R n , f and f are continuous functions.Assume that at least one subsystem is ISS, and without loss ofgenerality, suppose x-subsystem is ISS.Naturally, Lyapunov function is the most general choice forthe stability analysis of such system. However, it is not easyto find an appropriate candidate function, especially when thesystem becomes more and more complex and a lot of subsys-tems are strongly coupled with each other. If every subsystemis ISS, we can resort to the small-gain theorem for analysis,which uses the gain of each subsystem to check the stability ofthe interconnected system. Essentially, such a way representsa structural perspective and is more suitable for interconnectedsystems than the Lyapunov function based approach. Inspiredby this idea, this paper will propose a new structural stabilityanalysis approach for (5) and (6) in the following. Stability Analysis Procedure 1.
Step 1
Transform the interconnected form into a cascadeform via the ISS property of x-subsystem .The solution of x-subsystem can be written as a function ofthe initial value x ( t ) , t and input u ( t ) , i.e. x ( t )= φ ( x , t, z ) .Substituting this equation into the z-subsystem yields ˙ z = f ( z, φ ( x , t, z )) . Thus, the interconnected system becomesa cascade system of the following form ˙ z = f ( z, φ ( x , t, z )) (7) ˙ x = f ( x, z ) (8) Remark 1.
Using the ISS property, the above process hasan intuitive explanation. Since the x-subsystem is ISS w.r.t. z ( t ) and suppose the gain function from z ( t ) to x ( t ) is γ ,let γ ( z ( t )) be the input, then the x-subsystem corresponds toa filter whose gain is 1. Its output is not arbitrary but keepstracking γ ( z ( t )) , and in fact, it is kept in a neighboring areaof γ ( z ( t )) . Therefore, equation (8) can be written as the fol-lowing formulation of ISS x ( t ) = β ( x , t ) + γ ( z ( t )) + ∆ (9)where ∆ denotes a static error with γ ( z ( t )) whose specificform will be given later. Therefore, using (9), we can constructa feedback loop as follows ˙ z = f ( z, β ( x , t ) + γ ( z ) + ∆) (10)ig. 1: The cascade form of the interconnected systemand the cascade system can be described in the Fig.1. Step 2.
Analyze the stability of the feedback loop of z-subsystem.Since β ( x , t ) is convergent, the stability of feedback loopof z-subsystem depends on the function f and γ ( z ) + ∆ . Itis necessary to study the new formulation of ISS. Remark 2.
Compare with the original formulation of ISS, γ ( z ) + ∆ replaces γ ( sup τ ∈ [ t ,t ) k u ( τ ) k ) . It should note that theformer represents the current value of the input z ( t ) ,but thelater represents its history. Besides, in various proofs of thesmall-gain theorems([5]-[7]), only the form of the gain γ isneeded, but ∆ is not cared absolutely. But later, we will showwhat is ∆ and what does it function in the new problem. Step 3
Analyze the stability of the cascade system.After transformed into the cascade system, by the stabilitytheorems in [11] about cascade systems and the ISS property ofx-subsystem , if and only if the z-subsystem is stable, the entiresystem is stable, so is the original interconnected system.
Remark 3.
In this approach, the concept of ISS bridges thegap of the interconnected system and the cascade system. Inthis way, the stability of the complex interconnected system isequivalent to the stability of its equivalent cascade system, andfurther the stability of a feedback loop. Through transformingthe stability of the interconnected system into the stability ofone of subsystems, this approach greatly simplifies the analy-sis.It should be mentioned that this approach just requires onesubsystem should be ISS and need not construct an overallLyapunov function which used to consider all details and toodepends on the specific form of the system.
In this section, we present a new formulation of ISS con-cept to weaken the conservative of the original one. Considerthe general system (2) and suppose it satisfies the followingassumption.
Assumption 1.
The general system (2) satisfies the proposi-tion 1 and the gain function γ = α − ◦ α ◦ ρ is differentiable.Then we have the new formulation of ISS in the following. Theorem 1.
Suppose the system (2) satisfies the assumption1 in D , if there exist a class KL function β , a class K function γ , and a constant L > ,such that for any initial state x ( t ) and any bounded input u ( t ) , the solution of x ( t ) exists for all t ≥ t and satisfies k x k ≤ β ( x , u , t ) + γ ( k u k ) − L Z t e − R ts k ( τ ) dτ α ( u ) ˙ uds where α ( u ) = dα ( ρ ( k u k )) du and k ( t ) is continuous positivedefinite on R . proof. By the proposition 1, there exists the Lyapunov func-tion V ( x ) such that α ( k x k ) ≤ V ≤ α ( k x k ) (11) ˙ V ≤ − W ( V ) , ∀ k x k ≥ ρ ( k u k ) > . (12)where W is positive definite on R , and W ( V ) can be ob-tained by (11). According to (11), the condition of (12) k x k ≥ ρ ( k u k ) can be strengthened as V ≥ α ( ρ ( k u k )) , anddefine the error e = V − α ( ρ ( k u k )) ,then we obtain the errorsystem of equation (12) ˙ e ≤ − W ( e + α ( ρ ( k u k ))) − α ( u ) ˙ u, ∀ e ≥ (13)where α ( u ) = dα ( ρ ( k u k )) du . Since W is positive definite, wehave k ( t ) = W ( e + α ( ρ ( k u k ))) e > , ∀ e > . Especially, e = 0 means ˙ u = 0 and k x k ≤ γ ( k u k ) . Therefore, (13) can bewritten as ˙ e ≤ − k ( t ) e − α ( u ) ˙ u. (14)Solving it and by the comparison theorem in [10] yields e ( t ) ≤ e − R t k ( s ) ds e − Z t e − R ts k ( τ ) dτ α ( u ) ˙ uds (15)Due to R t k ( s ) ds > , we define the class KL function β ( x , u , t ) = e − R t k ( s ) ds e . In view of e = V − α ( ρ ( k u k )) ,equation (15) can be written as V ( t ) ≤ β ( x , u , t )+ α ( ρ ( k u k )) − Z t e − R ts k ( τ ) dτ α ( u ) ˙ uds. (16)That is k x k ≤ α − ( β ( x , u , t )+ α ( ρ ( k u k )) − Z t e − R ts k ( τ ) dτ α ( u ) ˙ uds ) . By the Lagrange median theorem, we have α − ( β ( x , u , t ) + α ( ρ ( k u k )) − Z t e − R ts k ( τ ) dτ α ( u ) ˙ uds ) − α − ( α ( ρ ( k u k ))= dα − ( x ) dx | x = ξ ( β ( x , u , t ) − Z t e − R ts k ( τ ) dτ α ( u ) ˙ uds ) (17)where x denotes α ( ρ ( k u k )) . Due to α − ( x ) ∈ K and dα − ( x ) dx > , for x ∈ D , there exists a constant L > ,such that k x k ≤ α − ( α ( ρ ( k u k )) + Lβ ( x , u , t ) − L Z t e − R ts k ( τ ) dτ α ( u ) ˙ uds. (18)t last, equation (18) can be written as k x k ≤ β ( x , u , t ) + γ ( k u k ) − L Z t e − R ts k ( τ ) dτ α ( u ) ˙ uds (19)where β = Lβ , γ = α − ◦ α ◦ ρ . Remark 4.
Compare with the old form, γ ( k u k ) − L R t e − R ts k ( τ ) dτ α ( u ) ˙ uds is more ac-curate than γ ( sup τ ∈ [ t ,t ) k u ( τ ) k ) . Recall equation(9), ∆ = − L R t e − R ts k ( τ ) dτ α ( u ) ˙ uds , and its convergencedepends on the existence of lim t →∞ R t e − R ts k ( τ ) dτ α ( u ) ˙ uds . Remark 5.
The new formulation reflects the tendency of x ( t ) tracking u ( t ) . If ignore the influence of x ( t ) , the neigh-boring area is determined by ∆ , that is when ˙ u → , ∆ → ,then k x ( t ) k ≤ γ ( k u k ) or k x ( t ) k → γ ( k u k ) . When ˙ u isbounded and ∆ is exists, x ( t ) keeps in a specific neighboringarea of u ( t ) . When ˙ u → ∞ or ∆ does not exists, the neigh-boring area is boundless. In other words, if we treat u ( t ) asthe leader and x ( t ) as the follower, α ( u ) ˙ u denotes the changerate of the leader, while e − R ts k ( τ ) dτ stands for the tracking rateof the follower.Based on the theorem 1, we have another formulation of ISSin the following. Corollary 1.
Suppose the condition of theorem 1 is satisfied,the solution of system (2) can be written as . k x k ≤ β ( x , u , t ) + α − ((1 − e − R tξ k ( s ) ds ) α ( ρ ( k u k ))) (20)where ξ ∈ [0 , t ] and k ( t ) is continuous positive definite on R . proof. By the integral median theorem, − R t e R ts k ( τ ) dτ α ( u ) ˙ uds in equation (15) can be writtenas − Z t e R ts k ( τ ) dτ α ( u ) ˙ uds = − e − R tξ k ( s ) ds Z t α ( u ) ˙ uds = − e − R tξ k ( s ) ds α ( ρ ( k u k ))+ e − R tξ k ( s ) ds α ( ρ ( k u ( t ) k )) (21)where ξ ∈ [0 ,t ] . So equation(16) can be written as V ≤ β ( x , u , t ) + α ( ρ ( k u k )) − e − R tξ k ( s ) ds α ( ρ ( k u k ))+ e − R tξ k ( s ) ds α ( ρ ( k u ( t ) k )) (22)Since e − R tξ k ( s ) ds ∈ (0 , , we have V ≤ β ( x , u , t ) + (1 − e − R tξ k ( s ) ds ) α ( ρ ( k u k )) (23)where β ( x , u , t )= β ( x , u , t ) + e − R tξ k ( s ) ds α ( ρ ( k u ( t ) k )) .Following the way of theorem 1, for () we obtain k x k ≤ β ( x , u , t ) + α − ((1 − e − R tξ k ( s ) ds ) α ( ρ ( k u k ))) Remark 6.
Compare two formulations (19) and (20), theformulation (19) containing ˙ u is suitable for the analysis of theconsensus of multi-agent systems, while (20) only containing u is convenient for the general stable system .The following simple example can illustrate the theorem 1.Now ,let us consider a simple case. Let u ∈ R in the system(2), we obtain the system that follows ˙ x = f ( x, u ) (24)It satisfies the assumption. Assumption 2.
System (24) is ISS, and there exists differen-tiable function γ : R n → R n whose element γ i belongs to theclass K function, such that f ( γ ( u ) , u ) (25) Remark 7.
The assumption represents a class of systemwhose equilibrium point is a class K function of input u ( t ) ,that means its static gain can be obtained by solving the alge-braic equation (25). There are many examples, e.g. all linearsystems and the following nonlinear systems ˙ x = − x + u and ˙ x = − tan x + tan u .Before moving on, we introduce a lemma first. Lemma 1.
If the function matrix A ( x ) = { a ij ( x ) } , a ij : R n → R , x ∈ D , ∀ i, j = 1 , ..., n , is negative (or positive)definite, there exist scalar functions λ ( x ) < λ ( x ) < (or λ ( x ) > λ ( x ) > )such that λ ( x ) I < A ( x ) < λ ( x ) I. Proof.
Since for any square matrix A = { a ij } , ∀ i, j = 1 , ..., n ,there exists a non-singular matrix P and a Jordan form J = diag ( J i ) , ∀ i = 1 , ..., m, m ≤ n ,such that A = P JP − .Then for A ( x ) , by the continuity, we have A ( x ) = P ( x ) J ( x ) P − ( x ) . (26)When A ( x ) is negative definite in D , its every engenvalue λ i,j ( x ) of J i is negative,where j = 1 , ..., rank ( J i ) .Then by the continuity, there exists λ min ( x ) < λ max ( x ) < such that λ min ( x ) < λ ij ( x ) < λ max ( x ) , x ∈ D , e.g.,we can let λ min ( s ) = inf s ≤k x k≤ r ( λ i ( x )) , ≤ s ≤ r , and λ max ( s ) = sup k x k≤ s ( λ i ( x )) , ≤ s ≤ r, ∀ i = 1 , ..., m , j =1 , ..., rank ( J i ) . It should be mentioned that if D = R n , r can be ignored.Thus, every diagonal element of J ( x ) − λ min ( x ) I can bewritten as ˆ λ ij ( x ) = λ ij ( x ) − λ min ( x ) > . By the propertyof the positive definite matrix , J ( x ) − λ min ( x ) I is positivedefinite , therefore, we have J ( x ) > λ min ( x ) I .Similarly, weobtain J ( x ) < λ max ( x ) I . Combing two inequalities yields λ min ( x ) I < J ( x ) < λ max ( x ) I (27)Multiply P ( x ) and P − ( x ) on the both sides of (27),and define λ ( x ) = λ min ( x ) and λ ( x ) = λ max ( x ) yields λ ( x ) I < ( x ) < λ ( x ) I Similarly, in the same way, we can obtainabove result when A ( x ) is positive definite.Based on above lemma, we have the following theorem. Theorem 2.
Suppose system (24) satisfies the assumption 2,if there exist class KL functions β i < β i , ∀ i = 1 , ..., n ,anda class K differentiable function γ such that for any initial state x ( t ) and any bounded input u ( t ) , the solution of x ( t ) existsfor all t ≥ t and satisfies ( x i ( t ) ≤ β i ( x , u , t ) + γ i ( u ) − R t e − R ts λ ( τ ) dτ min( γ ′ i ˙ u ) dsx i ( t ) ≥ β i ( x , u , t ) + γ i ( u ) − R t e − R ts λ ( τ ) dτ max( γ ′ i ˙ u ) ds (28)where i = 1 , ..., n , λ ( u, e ) ≥ λ ( u, e ) > . Proof.
Define the error e = x − γ ( u ) , then the error systemof system (24) can be written as ˙ e = f ( e + γ ( u ) , u ) − γ ′ ˙ u (29)where γ ′ = dγ ( u ) du .Expanding f ( e + γ ( u ) , u ) yields f ( e + γ ( u ) , u ) = f ( γ ( u ) , u ) + ∇ f ′ ( ξ + γ ( u ) , u ) e (30)where ξ = θe, θ ∈ [0 , . Due to f ( γ ( u ) , u )= 0 , then (29) canbe written as ˙ e = A ( u, e ) e − γ ′ ˙ u (31)where A ( u, e ) = ∇ f ′ ( ξ + γ ( u ) , u ) .Because when ˙ u = 0 , system (24) will converge to the con-stant u , ˙ e = f ( e + γ ( u ) , u ) is asympototiclly stable. By propo-sition 1, there exists Lyapunov function V ( e ) such that α ( k e k ) ≤ V ≤ α ( k e k ) (32) ˙ V = ∂V ( e ) ∂e A ( u, e ) e ≤ − W ( e ) , ∀ k e k ≥ ρ ( k ˙ u k ) (33)where α , α ∈ K ∞ and ρ ∈ K , W is a continuous positivedefinite function on R n .Calculating the derivative of (32) yields ∂α ( k e k ) ∂ k e k d k e k de ≤ ∂V ( e ) ∂e ≤ ∂α ( k e k ) ∂ k e k d k e k de (34)By the equivalence of the norms , there exists k , k > ,such that for any norm ,the following equation is true k e T e ≤k e k ≤ k e T e Calculating the derivative of above equationyields k e T ≤ d k e k de ≤ k e T (35)In view of (35), (34) can be written as ∂α ( k e k ) ∂ k e k k e T ≤ ∂V ( e ) ∂e ≤ ∂α ( k e k ) ∂ k e k k e T (36)Multiplying A ( u, e ) e to the both sides of (36) yields ∂α ( k e k ) ∂ k e k k e T A ( u, e ) e ≤ ∂V ( e ) ∂e A ( u, e ) e ≤ ∂α ( k e k ) ∂ k e k k e T A ( u, e ) e, (37) Consider (33), we obtain ∂α ( k e k ) ∂ k e k k e T A ( u, e ) e ≤ − W ( e ) . Due to α ∈ K , ∂α ( k e k ) ∂ k e k k > , then for any e ∈ R n , e T A ( u, e ) e < , that is A ( u, e ) is negative definite .By Lemma 1, there exist scalar functions λ ( u, e ) >λ ( u, e ) > such that − λ ( u, e ) I < A ( u, e ) < − λ ( u, e ) I .Then there exists derivative inclusive ˙ e = A s ( u, e ) e − b s ( u, ˙ u ) (38)where A s ( u, e ) ∈ {− λ ( u, e ) I, − λ ( u, e ) I } , b s ( u, ˙ u ) ∈{ min( γ ′ i ˙ u ) , max( γ ′ i ˙ u ) } .Solving (38) yields e i ( t ) ≤ e − R t λ ( u,s ) ds e i − Z t e − R ts λ ( u,e ) dτ min( γ ′ i ˙ u ) dse i ( t ) ≥ e − R t λ ( u,s ) ds e i − Z t e − R ts λ ( u,e ) dτ max( γ ′ i ˙ u ) ds In view of e = x − γ ( u ) and define β i ( x , u , t ) = e − R t λ ( u,e ) ds e i and β i ( x , u , t ) = e − R t λ ( u,e ) ds e i ,we obtain x i ( t ) ≤ β i ( x , u , t ) + γ i ( u ) − Z t e − R ts λ ( τ ) dτ min( γ ′ i ˙ u ) dsx i ( t ) ≥ β i ( x , u , t ) + γ i ( u ) − Z t e − R ts λ ( τ ) dτ max( γ ′ i ˙ u ) ds ∀ i = 1 , ..., n Remark 8 . Since γ ( u ) is the equilibrium point of (24), x ( t ) can converge towards γ ( u ) at last when ˙ u → , which isdifferent with theorem 1. The following two cases should benoted. If x ∈ R , (28) can be written as x ( t ) = β ( x , u , t ) + γ ( u ) − Z t e − R ts k ( τ ) dτ γ ′ ˙ uds (39)If γ i ( u ) = γ j ( u ) , ∀ i, j = 1 , ..., n ,(28) can be written as ( x i ( t ) ≤ β i ( x , u , t ) + γ i ( u ) − R t e − R ts λ ( τ ) dτ γ ′ i ˙ udsx i ( t ) ≥ β i ( x , u , t ) + γ i ( u ) − R t e − R ts λ ( τ ) dτ γ ′ i ˙ uds Example 3.
Some simple examples are ˙ x = − tan x + u and ˙ x = − x + 3 x + u ˙ x = − x + 2 x (40)If let system (24) be a scalar system, we have ˙ x = f ( x, u ) (41)where x ∈ R , f : R × R → R is the continuous and localLipschitz function w.r.t. x and u .Then like the corollary 1,we have the similar result. Corollary 2 . Suppose the condition of theorem 2 is satisfied,the solution of system (41) can be written as x ( t )= β ( x , u , t ) + (1 − e − R tξ a ( s ) ds ) γ ( u ( t )) (42)where ξ ∈ [0 , t ] , a ( x ) > , and β ∈ KL . Proof.
Based on (39) and following the way of Corollary 1we can prove the corollary 2.
Stability Analysis of Interconnected Systems
In this section, we analyze three typical kinds of intercon-nected systems via the approach proposed in the section III.
Consider the following systemx-subsystem: ˙ x = f ( x, z ) (43)z-subsystem: ˙ z = f ( z, x ) (44)where x ∈ R n , z ∈ R n , f and f are continuous functionssimilar with (2).x-subsystem and z-subsystem satisfy the proposition 1 andthe following assumption. Assumption 3.
The gain γ x of x-subsystem is defferentiable. γ x and the gain γ z of z-subsystem satisfies γ z ◦ γ x ( s ) < s, ∀ s ∈ [0 , ∞ ] . Then we have the following theorem.
Theorem 3.
If the nonlinear system (43)(44) are ISS andsatisfy the assumption 3. Then the interconnected system isasymptotically stable.
Proof.
Step 1.
Transform the interconnected system into thecascade system.According to proposition 1, for z-subsystem there exists theLyapunov function V ( z ) such that α z ( k z k ) ≤ V ≤ α z ( k z k ) (45) ˙ V ≤ − W ( z ) , ∀ k z k ≥ ρ z ( k x k ) where α z , α z ∈ K ∞ , ρ z ∈ K , and W is the continuouspositive definite function.Consider equation (45), the condition k z k ≥ ρ z ( k x k ) canbe strengthened as V ≥ α z ( ρ z ( k x k )) .For the x-subsystem, by corollary 1, we have k x k ≤ β ( x , z , t ) + α − x ((1 − e − R tξ k ( s ) ds ) α x ( ρ x ( k z k ))) . (46)where k ( t ) > .Substituting above equation into V ≥ α z ( ρ ( k x k )) yields V ≥ α z ( ρ z ( β ( x , z , t )+ α − x ((1 − e − R tξ k ( s ) ds ) α x ( ρ x ( k z k )))) Consider equation (45), above equation is strengthened as k z k ≥ α − z ◦ α z ◦ ρ z ( β ( x , z , t )+ α − x ((1 − e − R tξ k ( s ) ds ) α x ( ρ x ( k z k ))) , (47)Due to β ( x , z , t ) ∈ KL ,(47) is equivalent to k z k > α − ◦ α ◦ ρ ( α − ((1 − e − R tξ k ( s ) ds ) α ( ρ ( k z k ))) (48)Due to (1 − e − R tξ k ( s ) ds ) ∈ (0 , and by the monotoni-cally increasing property of class K function, we have α − z ◦ α z ◦ ρ z ( α − x ( α x ( ρ x ( k z k ))) > α − z ◦ α z ◦ ρ z ( α − x ((1 − e − R tξ k ( s ) ds ) α x ( ρ x ( k z k ))) . That is γ z ◦ γ x ( k z k ) > α − z ◦ α z ◦ ρ z ( α − x ((1 − e − R tξ k ( s ) ds ) α x ( ρ x ( k z k ))) . So ˙ V ≤− W ( z ) , ∀ k z k > γ z ◦ γ x ( k z k ) .Then the original interconnected system is transformed intothe following cascade form (cid:26) ˙ V ≤ − W ( z ) , ∀ k z k > γ z ◦ γ x ( k z k )˙ x = f ( x, z ) (49) Step 2.
Analyze the stability of the feedback loop.By the assumption 2, there exists k z k > γ z ◦ γ x ( k z k ) , suchthat z-subsystem is stable . Step 3 . Analyze the stability of cascade system.By the lemma 4.7 in [10] and the ISS property of x-subsystem , the cascade system (49) is stable, so is the inter-connected system (43)(44).
If at least one of subsystems in an interconnected system isnot ISS, the small-gain theorem is not used directly, e.g., thefollowing system.x-subsystem: ˙ x = f ( x, − z ) (50)z-subsystem: ˙ z = f ( t, x ) (51)where x ∈ R , z ∈ R , f and f are continuous functions,x-subsystem is ISS and z-subsystem satisfies the following as-sumption Assumption 4. ˙ z = f ( t, − k ( t ) ρ ( z )) is stable , where z ∈ D , > k ( t ) > and γ ∈ K is the differentiable gain functionof x-subsystem .There exists the following theorem. Theorem 4.
If the nonlinear system (50)is ISS and (51)satisfies the assumption 4. Then the interconnected system isasymptotically stable in D . Proof.
Step 1.
Transform the interconnected system into thecascade system.By corollary 2, equation (50) can be written as x ( t )= β ( x , u , t ) − (1 − e − R tξ a ( s ) ds ) γ ( z ( t )) (52)where γ ∈ K is the gain of (50)and a ( t ) > . Taking theLyapunov function V ( z ) for (51) yields ˙ V = ∂V∂z f ( t, x ) (53)Substituting (52) into (53) yields the cascade system as follows (cid:26) ˙ V = ∂V∂z f ( t, β ( x , u , t ) − (1 − e − R tξ a ( s ) ds ) γ ( z ( t )))˙ x = f ( x, − z ) (54) Step 2.
Analyze the stability of the feedback loop.By the theorem 3.4 in [10] , ignore β ( x , u , t ) , we have ˙ V = ∂V∂z f ( t, − (1 − e − R tξ a ( s ) ds ) γ ( z ( t ))) . By the assumption3 and > − e − R tξ a ( s ) ds > , z-subsystem is stable. Step 3.
Analyze the stability of cascade system.y the lemma 4.7 in [10] and the ISS property of x-subsystem , the cascade system (54) is stable, so is the inter-connected system (50)(51).
Example 4.
Other simple examples are in the following. (cid:26) ˙ x = x ˙ x = − (1 + x ) x − x and (cid:26) ˙ x = − x + x ˙ x = − x Traditionally, Lyapunov function based method is the popu-lar tool to analyze the consensus. Here, we will present anotherway from the structural perspective.In order to illustrate how touse the method proposed in section III to analyze the consen-sus problem, we consider the following nonlinear multi-agentsystem .x-subsystem: ˙ x = ( z − x ) (55)z-subsystem: ˙ z = ( x − z ) (56)where z, x ∈ R ,suppose γ x and γ z are the gains of the x-subsystem and z-subsystem respectively.Even though every system is ISS, the composition of thegain γ x ( s ) ◦ γ z ( s ) = s causes the small-gain theorem is notapplied directly in the situation. Following the method pro-posed in this paper, we have the following steps. Step 1.
Transform the interconnected system into the cas-cade system.By the theorem 2, the solution of x-subsystem is x ( t ) = β ( x , u , t ) + z − Z t e − R ts a ( τ ) dτ ˙ zds where a ( t ) > . Substituting above equation into z-subsystemyields ˙ z = L ( z, t )( β ( x , u , t ) − Z t e − R ts a ( τ ) dτ ˙ zds ) (57)where L ( z, t ) = ( β ( x , u , t ) − R t e − R ts a ( τ ) dτ ˙ zds ) . Thenthe original system is transformed into the following cascadesystem ˙ z = L ( z, t ) β ( x , u , t ) − L ( z, t ) Z t e − R ts a ( τ ) dτ ˙ zds (58) ˙ x = ( z − x ) (59) Step 2.
Analyze the stability of the feedback loop.For z-subsystem, define q ( t ) = e R t a ( s ) ds e R t L ( z,s ) ds Mul-tiplying both sides of (58) by q ( t ) and rearrange the equationwe obtain q ( t ) ˙ z + q ( t ) L ( z, t ) Z t e − R ts a ( τ ) dτ ˙ zds = q ( t ) L ( z, t ) β ( x , u , t ) , (60) From the expression of q ( t ) , one can verify q ( t ) ˙ z + q ( t ) L ( z, t ) Z t e − R ts a ( τ ) dτ ˙ zds = ddt ( e R t L ( z,s ) ds Z t e R s a ( τ ) dτ ˙ zds ) (61)Using (61) in (60) and integrating both sides of (60), we obtain e R t L ( z,s ) ds R t e R s a ( τ ) dτ ˙ zds = R t q ( s ) L ( z, s ) β ( x , u , s ) ds .Therefore, Z t e R s a ( τ ) dτ ˙ zds = Z t e R s a ( τ ) dτ e R ts − L ( z,τ ) dτ L ( z, s ) β ( x , u , s ) ds By theorem 2,due to β ( x , u , t ) = e − R t a ( s ) ds ,then R t e R s a ( τ ) dτ ˙ zds = R t e R ts − L ( z,τ ) dτ L ( z, s ) ds. We obtain ˙ z = L ( z, t ) β ( x , u , t ) − L ( z, t ) e − R t a ( s ) ds Z t e R ts − L ( z,τ ) dτ L ( z, s ) ds. Since L ( z, s ) = dds ( R ts − L ( z, τ ) dτ ) , we have Z t e R ts − L ( z,τ ) dτ L ( z, s ) ds = e R ts − L ( z,τ ) dτ (cid:12)(cid:12) t = 1 − e R t − L ( z,τ ) dτ . Due to e R ts − L ( z,τ ) dτ ∈ KL , we have R t e R ts − L ( z,τ ) dτ L ( z, s ) ds → , and e − R t a ( s ) ds R t e R ts − L ( z,τ ) dτ L ( z, s ) → , ∀ t → ∞ . Namely, R t e − R ts a ( τ ) dτ ˙ zds → , and L → , ∀ t → ∞ .Thus ˙ z → , ∀ t → ∞ . Because L ( z, t ) β ( x , u , t ) > , R t L ( z, t ) β ( x , u , t ) ds is integrable, so is R t L ( z, t ) e − R t a ( s ) ds R t e R ts − L ( z,τ ) dτ L ( z, s ) ds , therefore z ( t ) = z (0) + t Z L ( z, t ) β ( x , u , t ) ds − L ( z, t ) e − R t a ( s ) ds Z t e R ts − L ( z,τ ) dτ L ( z, s ) ds (62)exists. Step 3.
Analyze the stability of cascade system.By the theorem 2, for x-subsystem , when z ( t ) → c where c is a constant, x ( t ) → z ( t ) , ∀ t → ∞ . Therefore, the system isconsensus. In this paper, based on the structural perspective, we pro-pose a new framework for the stability analysis of intercon-nected systems. This method bases on a new formulation ofISS concept which weakens the conservative of the originalform and can analyze some problems which can not be doneby the small-gain theorem. As its applications, we also investi-gates the three kinds of typical interconnected systems respec-tively. It should be mentioned that we use a unifying approacho treat three different problems and provide a deep insight forthe common essence of them. Further more this method canbe used to design the lower-order controller for the high-orderminimum-phase systems e.g.[13][14][15]. In the future, wewill combine this method and the idea in [12] to analyze theconsensus of MAS with nonlinear protocol on any topologyand with the time-delay in the communication.
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