Neighbourhood conditions for network stability with link uncertainty
Simone Mariano Michael Cantoni


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§ ABSTRACT 

The main result relates to structured robust stability analysis of an input-output model for networks with link uncertainty.

It constitutes a collection of integral quadratic constraints, which together imply robust stability of the uncertain networked dynamics. Each condition is decentralized in the sense that it depends on

model data pertaining to 


the neighbourhood of 

a specific agent. By contrast, pre-existing conditions for the network model are link-wise decentralized, with each involving conservatively more localized problem data. A numerical example is presented to 
illustrate the advantage of the new broader neighbourhood conditions.  










Integral-Quadratic Constraints (IQCs), Network Robustness, Scalable Analysis.




§ INTRODUCTION


Motivated by problems in power and water distribution, transportation, ecology, and economics, 

large-scale networks of dynamical systems have been long studied in system and control theoretic terms; e.g., see <cit.> for state-space methods, and <cit.> for input-output methods. 

In this paper, an input-output approach, based on 


integral quadratic constraints (IQCs)


 <cit.>, is pursued to 

progress the 
development of 

scalable stability certificates for networks with uncertain links. The uncertain network model considered here was recently developed in <cit.>, with a focus on assessing the impact of link uncertainty expressed relative to the ideal (unity gain) link.  

The work is related to <cit.>, and as in <cit.>, most closely to <cit.>. The main contribution is 

an alternative approach to the decomposition of a monolithic IQC certificate that implies robust network stability. 


The new decomposition 


involves a 
collection of sufficient conditions, each depending on model data pertaining to a specific agent, its neighbours, and the corresponding links. That is, each condition is local to a specific neighbourhood. As such, compared to the link-wise decomposition 

presented in <cit.>, each neighbourhood based condition involves more (still localized) model data. This provides scope for reduced conservativeness, as illustrated by a numerical example.



The paper is organized as follows: Various preliminaries are established next. In Section <ref>, the structured feedback model of a network with uncertain links, and a correspondingly structured IQC based robust stability condition, which is amenable to decomposition, are recalled from <cit.>. Then, in 

Section <ref>, 
the novel neighborhood based decomposition is developed, alongside statement of the link-wise conditions from <cit.> for comparison, including a numerical example. Some concluding remarks are provided in Section <ref>.



§ PRELIMINARIES
 



 §.§ Basic notation
 
The natural, real, and complex numbers are denoted , , and , respectively.

For i,j∈, [i:j]:={k∈ | i≤ k ≤ j}, which is empty if j<i, _∙α:={β∈ℝ | β∙α} for given order relation ∙∈{>,≥,<,≤}, and [α,β]:=ℝ_≥α∩ℝ_≤β. 

With ∈{,}, for p∈, the p-dimensional Euclidean space over  is denoted by ^p. Given x∈^p, for i∈[1:p], the scalar x_i∈ denotes the i-th coordinate.


The vectors 1_p,0_p∈ℝ^p satisfy 1=(1_p)_i=1+(0_p)_i for every i∈[1:p].

^p× q denotes the space of p× q matrices over , for p,q∈.

The identity matrix is denoted by I_p ∈𝔽^p× p, the square zero matrix by O_p∈𝔽^p× p, and the respective p× q matrices of ones and zeros by 1_p× q and 0_p× q. Given M∈^p× q, for i∈[1:p], j∈[1:q], the respective matrices M_(·,j)∈^p× 1∼^p, and M_(i,·)∈^1× q, denote the j-th column, and i-th row. Further, M_(i,j)∈ denotes the entry in position (i,j). Given x∈𝔽^p, the matrix M=diag(x)∈𝔽^p× p is such that M_(i,i)=x_i and M_(i,j)=0, j≠ i∈[1:p], whereby I_p=diag(1_p). The transpose of M∈𝔽^p× q is denoted by M^'∈𝔽^q× p, and M^*=M̅^' denotes the complex conjugate transpose. For M_i=M_i^*∈𝔽^p× p, i∈{1,2}, M_i≻ 0 means there exists ϵ>0 such that x^* M_i x ≥ϵ x^*x for all x∈𝔽^p, M_i≽ 0 means x^* M_i x ≥ 0 for all x∈𝔽^p, and M_1≽ (resp. ≻) M_2 means M_1-M_2 ≽ (resp. ≻) 0. 



 §.§ Signals and systems

The Hilbert space of square integrable signals v=(t∈_≥ 0↦ v(t)∈^p) is denoted Ł_2 ^p, where the inner-product ⟨ v, u ⟩ := ∫_0^∞v(t)^' u(t) t and norm v_2 := ⟨ v, v ⟩^1/2 are finite; the superscript is dropped when p=1. The corresponding extended space of locally square integrable signals is denoted by Ł_2e^p; i.e.,  v:_≥ 0→^p such that π_τ(v)∈Ł_2 ^p for all τ∈_≥ 0, where (π_τ(v))(t):=f(t) for t∈[0,τ), and (π_τ(v))(t):=0 otherwise. 


The composition of maps F:Ł_2e^p ↦Ł_2e^r and G:Ł_2e^q ↦Ł_2e^p is denoted by F∘ G := (v↦ F(G(v)), and the direct sum by F⊕ G:=((u,v)↦ (F(u),G(v))). Similarly, ⊕_i=1^n G_i = G_1⊕⋯⊕ G_n. When G is linear, in the sense (∀α,β∈) (∀ u,v∈Ł_2e^q) G(α u + β v) = α G(u) + β G(v), the image of v under G is often written Gv, and in composition with another linear map ∘ is dropped. The action of a linear G:Ł_2e^q→Ł_2e^p corresponds to the action of p· q scalar systems G_(i,j):Ł_2e→Ł_2e, i∈[1:p], j∈[1:q], on the coordinates of the signal vector input associated with  Ł_2e^q∼Ł_2e×⋯×Ł_2e; i.e., (Gv)_i = ∑_j=1^q G_(i,j)v_j. Matrix notation is used to denote this. Also, for convenience, the map of pointwise multiplication by a matrix on Ł_2e is not distinguished in notation from the matrix. 

A system is any map G:Ł_2e^q→Ł_2e^p, with G(0)=0, that is causal in the sense  π_τ(G(u)) = π_τ(G (π_τ(u)) for all τ∈_≥ 0. It is

called stable if u∈Ł_2 ^q implies G(u) ∈Ł_2 ^p and  G:=sup_0≠ uG(u)_2/u_2< ∞ (the composition of stable systems is therefore stable.) The feedback interconnection of G with system Δ:Ł_2e^p →Ł_2e^q is well-posed if for all (d_y,d_u)∈Ł_2e^p×Ł_2e^q, there exists unique (y,u)∈Ł_2e^p×Ł_2e^q, such that 

    y = G(u) + d_y,      u = Δ(y) + d_u,

and [[G,Δ]] := ((d_y,d_u) ↦ (y,u)) is causal; see Figure <ref>. If, in addition, [[G,Δ]] < ∞, then the closed-loop is called stable. 


The following result is the well-known IQC robust feedback stability theorem, taken from <cit.>:


Given stable system Δ:Ł_2e^p →Ł_2e^q, and bounded linear map Π:Ł_2 ^p×Ł_2 ^q →Ł_2 ^p×Ł_2 ^q that is self-adjoint in the sense (∀ g_1,g_2∈Ł_2 ^p×Ł_2 ^q) ⟨ g_1, Π g_2 ⟩=⟨Π g_1, g_2⟩, suppose 

    ⟨
    
    
    (y,u),
    Π 
    (y,u)
    
    
    
    ⟩≥ 0,
      
    u=αΔ(y),


for all (y,α)∈Ł_2 ^p × [0,1]. Further, given stable system G:Ł_2e^q→Ł_2e^p, suppose [[G,αΔ]] is well-posed for all α∈[0,1], and there exists ϵ>0 such that 

    ⟨
    
    
    (y,u),
    Π 
    (y,u)
    
    
    
    ⟩≤ -ϵ u_2^2,
      
    y = G(u),

for all u∈Ł_2 ^q.
Then, GΔ is stable. 





 §.§ Graphs
 
Let =(,) be a simple (self-loopless and undirected) graph, where =[1:n] is the set of n∈ℕ∖{1} vertices, and ⊂{{i,j} | i,j∈} is the set of m:=||∈ℕ edges. Bijective κ_:→ with :=[1:m] denotes a fixed enumeration of the edge set  for indexing. The set _i:={j | {i,j}∈} comprises the neighbours of i∈, _i:={{i,j} | j∈𝒩_i} is the corresponding neighbourhood edge set, 
and bijective κ__i:_i→_i is a fixed enumeration of the m_i:=|_i| edges, where _i:=[1:m_i]. The edge indexes associated with i∈ are gathered in the set denoted by _i:={κ_({i,j}) | j ∈_i}. For each k∈, the set _k:=(_i ∪_j) ∖{k}, where {i,j} = κ_^-1(k), is the collection of all indexes of the edges associated with either of the neighbouring vertices i and j, excluding the one linking them. The following technical result regarding the given graph 𝒢=(,) is used subsequently.

    
    Given arbitrary E_k,F_k:Ł_2 ^p →Ł_2 ^p 
    for k∈: 
    
    
    
    
    
        
  *  ∑_i∈∑_k∈_i12F_k= ∑_k∈ F_k;
        
  *  ∑_i∈∑_k∈_i ∑_ℓ∈_i∖{k} E_k ∘ F_ℓ = ∑_k∈∑_ℓ∈_k E_k ∘ F_ℓ.
        
    


See Appendix. 





§ NETWORKED SYSTEM MODEL AND ROBUST STABILITY ANALYSIS



In this section, the network model and robust stability analysis from <cit.> is recalled first. A new result is then derived to underpin the aforementioned neighbourhood decomposition, which is subsequently developed in Section <ref>. 

Consider a network of n∈∖{1} dynamic agents, coupled according to the simple graph =(,). The vertex set =[1:n] corresponds to a fixed enumeration of the agents, and  
m:=|| is the number of edges, defined according to {i,j}∈ if the output of agent i∈ is shared as an input to agent j∈, and vice-versa. It is assumed that the number of neighbours m_i:=|_i|≥ 1 for all i∈. Note that ∑_i∈ m_i=2m. To tame the notation, each agent has a single output and dynamics corresponding to the system H_i:Ł_2e^m_i→Ł_2e, which is taken to be linear and stable, with the vector input signal coordinate order fixed by the neighbourhood edge-set enumerations κ__i:_i→_i.

Define the block diagonal systems



    H:=⊕_i=1^nH_i:Ł_2e^2m→Ł_2e^n, 
    
       
    T:=⊕_i=1^n1_m_i× 1:Ł_2e^n→Ł_2e^2m, 
        and    R:=⊕_i=1^n(⊕_k=1^m_i R_i,k):Ł_2e^2m→Ł_2e^2m,


 where 1_m_i× 1:Ł_2e→Ł_2e^m_i denotes pointwise multiplication by 1_m_i× 1∈ℝ^m_i× 1, and the system R_i,k:Ł_2e→Ł_2e represents the stable but possibly nonlinear and time-varying dynamics of the link from agent i∈ to its neighbour j∈_i with {i,j}=κ__i^-1(k).  Given these components, the networked system can be modelled as the structured feedback interconnection [[P, R ∘ T ∘ H]], where P:Ł_2e^2m→Ł_2e^2m is pointwise multiplication by a permutation matrix arising from the structure of . More specifically, for each i∈, k∈_i, and r∈[1:2m], the corresponding entry of this permutation matrix is given by
 
    P_(∑_h∈ [1:i-1] m_h
     
     +k, r)= 1    if r=
     ∑_h∈ [1:j-1]m_h 
     
     
     +κ__j({i,j})
     
       with  j∈κ__i^-1(k)∖{i},
    
     0    otherwise,

 where by convention the sum over an empty index set is zero.
 See <cit.> for more details about the model and its components.
 


The `routing' matrix P=P^'=P^-1 is the adjacency matrix of an undirected 1-regular sub-system graph 
 :=(,), with 2m vertices and m edges, corresponding to the disjoint union of the two-vertex subgraphs 𝒢[e] induced by each edge e∈ in the network graph 𝒢=(,). As such, 

    P=D-L=I_2m-
     ∑_k∈ L_k,

where the degree matrix D=I_2m since  is 1-regular, and the Laplacian decomposes as
 
    L=∑_k∈L_k
      =∑_k∈B_(·,k)B_(·,k)^'
      =B B^',

 where B∈ℝ^2m× m is the incidence matrix
 defined by 

B_(r,k)=1=-B_(s,k) for {r,s}=κ^-1_(k), and B_(t,k):=0 for each t ∈ [1:2m]∖{r,s}, over k∈; the enumeration κ_ is taken to be compatible with the enumerations κ__i and the definition of P in (<ref>). The edge orientation is arbitrary.
 
 
 
 
 
 The network with ideal links (i.e., R=I_2m) is stable in the sense that [[P, T∘ H]] 
 is stable.
 
 






 





 













 


 










 


 






 

 





 


 









 




























































 


 










 


 










 


 







 













 


    








 

 
    


 











 

 









As detailed in <cit.>, one can leverage Assumption <ref> in the analysis of [[P,R∘ T ∘ H]] by considering uncertainty in the links R relative to ideal unity gain links.

Indeed, as illustrated in Figure <ref>, to verify robust stability of the networked system [[P,R∘ T ∘ H]] it is sufficient to verify that [[G,Δ]] is stable, where  

    Δ:= (R-I_2m) ∘ T,

and

    G := 
    
    H ∘ (P - T∘ H )^-1,


with the stable systems R, H, and T as per (<ref>), and P as per (<ref>). Since (P - T∘ H )^-1 is stable by Lemma 1 in <cit.>, G:Ł_2e^2m→Ł_2e^n, and  the uncertain Δ:Ł_2e^n→Ł_2e^2m, are both stable. 

 (<cit.>)

 
Under Assumption <ref>, if 

[[G,Δ]] is stable, with G as per (<ref>), and Δ as per (<ref>), then the networked system model [[P,R∘ T ∘ H]] is stable. Further, when Δ is also linear, stability of [[P,R∘ T ∘ H]] implies stability of [[G,Δ]].  




Suppose, for all (y,α)∈Ł_2 ^n× [0,1],

    (y,u)Φ(y,u)≥ 0,    u=αΔ(y),

where the given bounded linear self-adjoint operator Φ:(Ł_2 ^n×Ł_2 ^2m) → (Ł_2 ^n×Ł_2 ^2m) is structured according to

    Φ = [   Φ_1   Φ_2; Φ_2^*   Φ_3 ]
        =[  ⊕_i=1^n Φ_1,i  ⊕_i=1^n Φ_2,i; ⊕_i=1^nΦ_2,i^*  ⊕_i=1^n Φ_3,i ];

the superscript * denotes Hilbert adjoint. For example, this IQC holds by selecting each


    Φ_i := 
    [   Φ_1,i   Φ_2,i; Φ_2,i^*   Φ_3,i ]
    :Ł_2 ×Ł_2 ^m_i→Ł_2 ×Ł_2 ^m_i,   i∈,


such that (y_i,u_i)Φ_i(y_i,u_i)≥ 0, u_i=αΔ_i(y_i), for all (y_i,α)∈Ł_2× [0,1], with local Δ_i=(⊕_k=1^m_iR_i,k-I_m_i)∘1_m_i× 1. Then, by Theorem <ref>, the stability of GΔ is implied by the existence of ϵ>0 such that for all z∈Ł_2 ^2m,
 

    [ N; M ]z[   Φ_1   Φ_2; Φ_2^*   Φ_3 ][ N; M ]z≤ -ϵ z _2^2,
 
where

    N:= H,   and      M:=(P-T∘ H)=I_2m-L-T∘ H,

with T, H, and P, as per (<ref>) and (<ref>).
Importantly, N and M are structured coprime factors of G=NM^-1, 

as discussed in <cit.>. Also refer to <cit.> for more on the use of Theorem <ref> to arrive at (<ref>), and discussion of the obstacle to direct application of the decomposition method proposed in <cit.> to the 




equivalent condition

    [ I_2m;    L;      ]z[ [   Ξ_1   Ξ_2; Ξ_2^*   Ξ_3 ]] 
            [ I_2m;    L;      ]z≤ -ϵ z _2^2,


where L=∑_k∈ [1:m] L_k is the sub-system graph Laplacian in (<ref>),

    
    
    Ξ_1   := N^*Φ_1N+ N^*Φ_2 J+J^*Φ_2^*N+J^*Φ_3 J ,
    Ξ_2   :=-N^*Φ_2-J^*Φ_3 ,
    Ξ_3   :=Φ_3 ,

and J:= I_2m-T∘ H.

The block-diagonal structure of N, J, Φ_1, Φ_2, Φ_3, and the Hilbert adjoints when restricted to Ł_2, is such that Ξ_1 =  ⊕_i=1^n Ξ_1,i, Ξ_2= ⊕_i=1^n Ξ_2,i, and Ξ_3 =⊕_i=1^n Ξ_3,i, where 

    Ξ_1,i   := H_i^*Φ_1,iH_i +(I_m_i - 1_m_iH_i)^*Φ_3,i (I_m_i - 1_m_iH_i) 
          + H_i^*Φ_2,i(I_m_i - 1_m_iH_i) +(I_m_i - 1_m_iH_i)^*Φ_2,i^*H_i ,
    Ξ_2,i   :=-H_i^*Φ_2,i-(I_m_i-1_m_iH_i)^*Φ_3,i ,
    Ξ_3,i   :=Φ_3,i .


For context, an existing

link-wise
decomposition of (<ref>) is


first recalled from <cit.>.

 (<cit.>)  
Let W=∑_k∈ W_k ∈ℝ^2m× 2m be such that W_k≽ 0 and W≻ 0.
Suppose there exist X_k=X^*_k:Ł_2 ^2m→Ł_2 ^2m,  Z_k=Z_k^*:Ł_2 ^2m→Ł_2 ^2m, and ϵ_k>0, k∈, such that for all z∈Ł_2 ^2m,

    [ I_2m;  L_k;      ] z[ X_k+ϵ_k W_k         Ξ_2;       Ξ_2^*         Z_k ][ I_2m;  L_k;      ]z≤ 0,  k∈,
       zΞ_1 z - ∑_k∈zX_k z≤ 0,
       L zΞ_3 L z - ∑_k∈L_k zZ_k L_k z≤ 0,
  
with L_k as per (<ref>).
 Then, there exists ϵ>0 such that (<ref>) for all z∈Ł_2 ^2m.
 

The m=|| conditions in (<ref>) 

enable use of the structure of each L_k=B_(·,k)B^'_(·,k) for link-wise decentralized verification of (<ref>), although this can be conservative. 


An alternative is to decompose according to the broader structure of each

 
    K_i=∑_k∈_i L_k,    i∈,

 which encompasses the neighborhood of agent i.
 
 
 

Let W=∑_i∈ W_i ∈ℝ^2m× 2m be such that W_i≽ 0 and W≻ 0.
Suppose there exist X_i=X^*_i:Ł_2 ^2m→Ł_2 ^2m, Y_i:Ł_2^2m→Ł_2^2m, Z_i=Z_i^*:Ł_2 ^2m→Ł_2 ^2m, and ϵ_i>0, i∈, such that for all z∈Ł_2 ^2m,



    [ I_2m;  K_i;      ] z[ X_i+ϵ_i W_i         Y_i;       Y_i^*         Z_i ][ I_2m;  K_i;      ]z≤ 0,   i∈,
       zΞ_1 z - ∑_i∈zX_i z≤ 0,
       zΞ_2 L z - ∑_i∈zY_i K_i z≤ 0,
       L zΞ_3 L z - ∑_i∈K_i zZ_i K_i z≤ 0,
   

with K_i as per (<ref>).
 Then, there exists ϵ∈ℝ_>0 such that (<ref>) for all z∈Ł_2 ^2m.
 
 
	 For all z∈Ł_2^2m,   (<ref>) implies
	    
    zX_i z
             + zY_i K_i z + K_i zY_i^* z
             + K_i zZ_i K_i z≤ -ϵ_i zW_i z,

     for each i∈, and therefore, 
	    
    ∑_i∈ ( zX_i z   + zY_i  K_i z +  K_i zY_i^* z + K_i zZ_i K_i z ) 
       ≤ -∑_i∈ϵ_i zW_i z≤ - ϵ || z ||_2^2,

      where ϵ = (min_i∈ϵ_i) ·(min_x∈ℝ^n x^' W x/x^' x)>0; note that W=∑_i∈ W_i ≻ 0 implies ∑_i∈ϵ_i W_i ≽ (min_i∈ϵ_i) ∑_i∈  W_i≻ 0, because each W_i≽ 0.
	Combining (<ref>), (<ref>), (<ref>), and (<ref>), gives 
     
    zΞ_1 z + zΞ_2 L z  + L zΞ_2^*  z + L zΞ_3 L z≤ -ϵ || z ||_2^2,

 as claimed.
 
 

As with <cit.>, the proof of Lemma <ref> expands upon ideas from the proof of <cit.>, 
which is not directly applicable here for the reasons elaborated in <cit.>. Considering K_i as per Lemma <ref>, instead of L_k as per Lemma <ref>, makes each instance of (<ref>) depend on more network model data, which provides scope for reducing the conservativeness of Lemma <ref>. This comes at the cost of the additional inequality (<ref>),

which has no counterpart in Lemma <ref> since ∑_k∈ L_k = L, whereas ∑_i∈ K_i ≠ L. 





§ MAIN RESULT: NEIGHBOURHOOD CONDITIONS


A possible selection of W_i, X_i, Y_i, Z_i, i∈, is devised for Lemma <ref>. It yields a decentralized collection of conditions 

that together imply the stability of GΔ, 

and thus, stability of the network by Theorem <ref>.


The conditions are 
decentralized 

in the sense that each depends on model data that is local to the neighbourhood of a specific agent, including corresponding components of the IQC based uncertainty description of the local links.


The subsequent matrix definitions, and related properties, lead to the proposed selection of W_i, X_i Y_i and Z_i in Lemma <ref>. The definitions pertain to the structure of the networked system model, encoded by the network graph =(,) and corresponding sub-system graph =(,), with =[1:n], m=||=||,

||=2m=∑_i∈m_i, and m_i=|_i|, as per Section <ref>. First, for each k∈=[1:m], define

    A_k := (diag(B_(·,k)))^2 ∈ℝ^2m× 2m,

where B∈ℝ^2m× 2m is the incidence matrix of the sub-system graph Laplacian matrix L=∑_k∈ L_k = ∑_k∈ B_(·,k)B_(·,k)^' in (<ref>). The matrix A_k is diagonal, with {0,1} entries, and 

    (A_k)_(r,r)=1   if and only if   r∈κ^-1_(k);

i.e., the value is 1 only in the two locations corresponding to the sub-systems in  associated with link k,





as per the

definition of the incidence matrix B below (<ref>). Indeed, since =(,) is 1-regular, direct calculation gives A_kB_(·,k)=B_(·,k), B_(·,k)^' A_k = B_(·,k)^', A_k B_(·,ℓ)=0_2m, B_(·,ℓ)^' A_k=0_2m^',

    
    
    A_k L_k = (diag(B_(·,k)))^2 B_(·,k) B_(·,k)^' = 
            L_k = L_k^' = L_k A_k 
     and    A_k L_ℓ = (diag(B_(·,k)))^2 B_(·,ℓ) B_(·,ℓ)^' = O_2m = L_ℓ A_k

    for all k ≠ℓ∈. As such, given i∈, for k∈_i,
    
    (∑_ℓ∈_i L_ℓ)  A_k    = L_k =
            A_k (∑_ℓ∈_i L_ℓ),   and 
    
         ( ∑_k∈_i A_k ) ( ∑_ℓ∈_i L_ℓ )    = ∑_k∈_i L_k 
            = ( ∑_ℓ∈_i L_ℓ )  ( ∑_k∈_i A_k ),

     where the set _i of edge indexes associated with agent i is defined as per Section <ref>.
























Finally, 

for each i∈, define 


 
    C_i:=(T_(·,i)) ∈ℝ^2m× 2m,

where 

T is given in (<ref>). As such,

C_i=C_i^' is a diagonal matrix with {0,1} entries. 

Composing it with Ξ_1 in (<ref>)



isolates only the model data related to the agent i∈.

It can be shown by direct calculation that 
 
 
    C_i Ξ_1= Ξ_1 C_i, 
         and   ∑_i ∈ C_i=I_2m.

 

The following is used to prove the subsequent main result.


    For arbitrary linear = ⊕_i=1^n _i, _i:Ł_2^m_i→Ł_2^m_i,
    
    and the sub-system graph Laplacian L=∑_k∈ L_k in (<ref>), 
    
    
    L   L = ∑_k∈ L_k  L_k + ∑_k∈∑_ℓ∈_k L_k  L_ℓ ,

    where _k collects the edge indexes associated with the two agents linked by edge k, excluding the latter, as per the definition in Section <ref>.


See Appendix.



For each i∈, let 


 
    W_i    := C_i , 
    
        X_i    := C_i Ξ_1 = Ξ_1 C_i , 
    
        Y_i    :=  ∑_k∈_i12Ξ_2 A_k , 
    
        Z_i    := 
          ∑_k∈_i12  A_k Ξ_3 A_k + ∑_k∈_i ∑_ℓ∈_i ∖{k}A_k Ξ_3 A_ℓ ,
   

with A_k as per (<ref>), C_i as per (<ref>), and the linear block diagonal Ξ_1, Ξ_2, Ξ_3 as per (<ref>).


If for all i∈, there exists ϵ_i∈ℝ_>0 such that for all z∈Ł_2 ^2m,
  
    [ I_2m;  K_i;      ]z[ [ X_i+ϵ_i W_i         Y_i;       Y^*_i         Z_i ]][ I_2m;  K_i;      ] z≤0,

     with K_i as per (<ref>),
 then there exists ε∈ℝ_>0 such that (<ref>) for all z∈Ł_2 ^2m.


With Z_i as per (<ref>), 




    ∑_i∈K_i zZ_i K_i z
       =∑_i∈⟨  z, K_i  ( ∑_k∈_i12 A_k Ξ_3 A_k +∑_k∈_i∑_ℓ∈_i ∖{k}A_k Ξ_3 A_ℓ ) K_i z ⟩
    
    
       =∑_i∈⟨  z, (
     ∑_k∈_i12 L_k  Ξ_3 L_k  + ∑_k∈_i∑_ℓ∈_i ∖{k} L_k Ξ_3 L_ℓ)z
    ⟩
       = ⟨  z, (
    ∑_i∈∑_k∈_i12 L_k  Ξ_3 L_k  + ∑_i∈∑_k∈_i∑_ℓ∈_i ∖{k} L_k Ξ_3 L_ℓ) z
    ⟩
       = ⟨  z, ( ∑_k∈ L_k  Ξ_3 L_k  + ∑_k∈∑_ℓ∈_k L_k Ξ_3 L_ℓ) z ⟩
    
    =
    
    
    L zΞ_3 L z,

which implies (<ref>). The second equality holds by
the definition of K_i in (<ref>), 
and the identity

(<ref>), whereby (∑_l∈_i L_l) A_k = L_k and A_ℓ (∑_l∈_i L_l) = L_ℓ whenever k,ℓ∈_i.

Both parts of Lemma <ref> are used for the fourth equality, and Lemma <ref> for the final equality.

Similarly, with X_k as per (<ref>), given (<ref>), 


∑_i∈ X_i = ( ∑_i∈ C_i ) Ξ_1   = Ξ_1.

As such, ∑_i∈zX_k z = zΞ_1 z for all z∈Ł_2 ^2m, which implies (<ref>). Further, with 
Y_k as per (<ref>), in view of  the identity (<ref>), linearity of Ξ_2, and part <ref>) of Lemma <ref>, for all z∈ L_2^2m,

    ∑_i∈ Y_i K_i    = ∑_i∈12Ξ_2  (∑_k∈_iA_k)   (∑_ℓ∈_i L_ℓ ) 
       = 
    
     ∑_i∈12Ξ_2 ∑_k∈_i L_k  
    
       =
    Ξ_2 ∑_k∈ L_k,

which implies (<ref>).
Finally, with
W_i≽ 0 as per (<ref>), ∑_i∈  W_i = ∑_i∈  C_i = I_2m≻ 0; see (<ref>). As such, Lemma <ref> applies, and therefore, (<ref>) for all z∈Ł_2 ^2m, as claimed.



For comparison, the link-wise decomposition from <cit.> is recalled below.

 (<cit.>)
For each k∈[1:m], let 

    W_k    := A_k , 
    
        X_k    := 1/2(A_kΞ_1 + Ξ_1 A_k) , 
    
        Y_k    := Ξ_2A_k , 
    
        Z_k    := 
        A_k(⊕_i=1^n D_i)A_k ,

where A_k is defined in (<ref>), and diagonal D_i:Ł_2 ^m_i→Ł_2 ^m_i is such that Ξ_3,i=D_i+S_i for some negative semi-definite S_i:Ł_2 ^m_i→Ł_2 ^m_i, i∈, with Ξ_1, Ξ_2, Ξ_3 as per (<ref>). 

If for all k∈, there exists ϵ_k∈ℝ_>0 such that for all z∈Ł_2 ^2m
  
    [ I_2m;  L_k;      ]z[ [ X_k+ϵ_k W_k         Y_k;       Y^*_k         Z_k ]][ I_2m;  L_k;      ] z≤0,

 then there exists ε∈ℝ_>0 such that (<ref>) for all z∈Ł_2 ^2m.




In Proposition <ref>, Z_i is not restricted to depend on only a suitable diagonal component D_3 of Ξ_3. This is one aspect of Proposition <ref> that helps mitigate potential conservativeness of the link-based decomposition in Proposition <ref>. Further, the determination of X_i, Y_i, and Z_i in Proposition <ref> depends on model data available to the neighbors of agent i∈. More specifically, each depends on the dynamics of agent i, and its neighbours in _i, as well as the associated neighbourhood links corresponding to 

the index set _i.

This stems from the block-diagonal structure of the linear Ξ_p, p ∈ [1:3], in (<ref>), 

and the network structure in each component L_k, k∈, of the sub-system graph Laplacian L=∑_k∈L_k.

Indeed, the positions and values of the non-zero elements in K_i Z_i K_i reflect the nonzero pattern corresponding to each L_k for k∈_i, and the same for Y_i K_i; see the proof of Lemma <ref> for further detail. The definition of X_i, on the other hand, only depends on model data specific to agent i∈. This additional aspect of Proposition <ref> could potentially reduce the conservativeness of the decentralized conditions, compared to those in Proposition <ref>, as each of the latter relies on more localized information that pertains only to each link k∈.


Since 

the conditions in Proposition <ref> do not require the network to have any particular interconnection structure,

the 

result is generally applicable for the scalable robust stability analysis of sparsely interconnected large-scale systems.






To illustrate the scope for reduced conservativenss provided by Proposition <ref>, consider the following path graph network example with n=10 agents; i.e., m=9, m_1=m_10=1 and m_i=2 for i∈[2:9]. Suppose that the agent dynamics is such that the non-zero entries of H in (<ref>) are identical and linear time-invariant, with transfer function 1/(s+25). Further, suppose the links R are such that the block elements of the block diagonal Δ = (R-I_18)∘ T are sector bounded, with Φ_1,i=-2m_iαβ, Φ_2,i=1_m_i(α+β) and Φ_3,i=-2 I_m_i, with α=-2, β=0.15.


Linear Matrix Inequality (LMI) conditions for semi-definite programming-based verification of the monolithic IQC (<ref>), or (<ref>) in Proposition, <ref>, or verifying (<ref>) in Proposition <ref>, can be derived, respectively, via the well-known Kalman-Yakubovich-Popov (KYP) Lemma <cit.>, given a state-space model for the agent dynamics. The details have been omitted due to space limitations. The LMIs obtained for (<ref>) in Proposition <ref>, and those for (<ref>), are demonstrably feasible, thereby successfully guaranteeing the robust stability of the network. By contrast, the LMIs for (<ref>) in Proposition <ref> fail to be feasible for a suitable selection of parameters ϵ_k>0, k∈[1:m]. 





§ CONCLUSIONS



Neighbourhood decentralized robust stability conditions are devised for networked systems in the presence of link uncertainty. This result is based on input-output IQCs that are used to describe the link uncertainties, and ultimately the structured robust stability certificate. An example is used to illustrate the scope for reduced conservativeness compared to existing results. Future work will explore alternative decompositions and comparisons.

It is also of interest to apply the main result to study specific network scenarios where information exchange is impacted by asynchronous time-varying delays, and dynamic quantization (e.g., see <cit.>.)



Proof of Lemma <ref>. 

Item <ref>) holds because ⋃_i∈_i =, and for k∈, the cardinality of _k:={i∈ | k∈_i} is exactly 2, so that ∑_i∈∑_k∈_i1/2 F_k = ∑_k∈ (12F_k + 12F_k) = ∑_k∈ F_k.

Further, for all k∈, the edge index set 
    _k = ⋃_i∈_k_i∖{k},
 and (_i∖{k}) ⋂ (_j∖{k})=∅ whenever i, j ∈_k with i≠ j. Therefore, 

    ∑_i∈ ∑_k∈_i ∑_ℓ∈_i∖{k} E_k∘ F_ℓ    =
    ∑_k∈ ∑_i∈_k ∑_ℓ∈_i∖{k}  E_k∘ F_ℓ
       =
    ∑_k∈ ∑_ℓ∈_k  E_k∘ F_ℓ,

which is item <ref>).     □










   
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
    
    
 
    
    Proof of Lemma <ref>. 






















    
    First note that L_k   L_ℓ = O_2m for every k∈, and ℓ∈∖ (_k∪{k}).
    
    
    Indeed, given the definition of _k, and the block diagonal structure of = ⊕_i=1^n _i,
    if the sub-system graph vertex index r∈[1:2m] is such ( B_(·,ℓ))_r ≠ 0, then
    
    r ∉ [∑_h∈[1:i-1]m_h+1:∑_h∈[1:i]m_h]  ⋃  [∑_h∈[1:j-1]m_h+1:∑_h∈[1:j]m_h]

    with {i,j}=κ_^-1(k),
    whereby B_(r,k)=0. Therefore, (B_(·,k))^' ( B_(·,ℓ)) = 0, and thus, 
    L_k L_ℓ = B_(·,k)B_(·,k)^' B_(·,ℓ)B_(·,ℓ)^'=O_2m.
 

    Since  is linear by hypothesis, and pointwise multiplication by each L_k is linear, it follows from the preceding observation that
    
    L    L     = (∑_k∈ L_k)   (∑_ℓ∈ L_ℓ)
       = ∑_k∈ ∑_ℓ∈ L_k   L_ℓ
       = ∑_k∈ ∑_ℓ∈_k∪{k} L_k   L_ℓ
       = ∑_k∈ ( L_k   L_k + ∑_ℓ∈_k L_k    L_ℓ )

    as claimed.     □
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    




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