Distributed Synthesis of Local Controllers for Networked Systems with Arbitrary Interconnection Topologies
Etika Agarwal S. Sivaranjani Vijay Gupta Panos Antsaklis

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    December 30, 2023
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We consider the problem of designing distributed controllers to guarantee dissipativity of a networked system comprised of dynamically coupled subsystems. We require that the control synthesis is carried out locally at the subsystem-level, without explicit knowledge of the dynamics of other subsystems in the network. We solve this problem in two steps. First, we provide distributed subsystem-level dissipativity analysis conditions whose feasibility is sufficient to guarantee dissipativity of the networked system. 
We then use these conditions to synthesize controllers locally at the subsystem-level, using only the knowledge of the dynamics of that subsystem, and limited information about the dissipativity of the subsystems to which it is dynamically coupled. We show that the subsystem-level controllers synthesized in this manner are sufficient to guarantee dissipativity of the networked dynamical system. We also provide an approach to make this synthesis compositional, that is, when a new subsystem is added to an existing network, only the dynamics of the new subsystem, and information about the dissipativity of the subsystems in the existing network to which it is coupled are used to design a controller for the new subsystem, while guaranteeing dissipativity of the networked system including the new subsystem. Finally, we demonstrate the application of this synthesis in enabling plug-and-play operations of generators in a microgrid by extending our results to networked switched systems. 

 Distributed control synthesis, networked systems, distributed control, compositional control, microgrids, dissipativity. 


§ INTRODUCTION

Control of large-scale networked dynamical systems, comprising of several dynamically coupled subsystems, has recently gained prominence due to emerging applications in infrastructure networks. For example, in power networks, new architectures where subsystems comprised of small clusters of renewable generators and loads, known as microgrids, are connected to form the large-scale power grid have been proposed <cit.>. As another example, subsystems comprising of autonomous vehicles that communicate with each other to travel in close formation (platoons), have been proposed to alleviate congestion and enhance safety in transportation networks <cit.><cit.>. 

In such large-scale networks, decentralized and distributed control approaches to guarantee stability and robustness, have been proposed to decrease communication overhead and computational complexity. In these approaches, the subsystem-level controllers only use states from a subset of neighboring subsystems to determine their control actions, however, the process of designing the controllers is centralized, that is, the knowledge of the dynamics of all subsystems is used in the design <cit.>-wang1973stabilizationlau1972decentralizedaoki1971somebellman1974largedavison1990decentralized<cit.>. This centralized design process has two drawbacks. Firstly, it may be impractical to assume that the control designer has knowledge of the dynamics of all subsystems in the network. For example, in interconnected microgrid networks, where the internal dynamics of the microgrid are continually changing due to renewable energy and load fluctuations, it is not practical to access the dynamics of all the microgrids. Secondly, the topology and interconnection structure of a network may change due to addition or removal of subsystems. For example, in vehicle platooning applications, vehicles may enter or leave the platoon at any time. In such cases, a centralized design process will necessitate redesign of all controllers in the network, which is neither computationally scalable nor desirable for real-time operation.

Therefore, distributed synthesis, where controllers are designed locally at the subsystem-level without explicit knowledge of the dynamics of other subsystems, is the only viable option for the realization of such large-scale networks <cit.><cit.>. Typically, distributed synthesis of controllers has been carried out using three types of approaches. The first class of approaches relies on either exploiting or inducing weak coupling between subsystems in the network to distribute the synthesis problem <cit.>, <cit.>-sezer1986nestedsethi1998nearsivaranjani2017distributed<cit.>. The second class of approaches are based on using numerical techniques like methods of multipliers, subgradient algorithms or distributed invariant set computations to decompose the control synthesis problem into more tractable problems <cit.>-conte2012distributedzeilinger2013plugriverso2014plug<cit.>. The final class of approaches is hierarchical, involving a centralized computation of subsystem-level conditions to guarantee network-level control objectives such as stability, robustness or dissipativity, and local synthesis of controllers at the subsystem-level to guarantee these objectives <cit.>-ishizaki2019modularityxu2009distributedhudon2012dissipativitytan1990decentralized<cit.>. 

In this paper, we consider the problem of synthesizing distributed controllers to guarantee dissipativity  for a networked system comprised of dynamically coupled subsystems. We require that the controllers be designed locally at the subsystem-level without explicit knowledge of the dynamics of other subsystems in the network. The contributions of this paper in addressing this problem are as follows.

    
  * Distributed analysis: We first decompose a centralized dissipativity analysis condition on the networked dynamical system into conditions on the dissipativity of individual subsystems. Passivity analysis of a networked system comprised of dynamically coupled subsystems has been studied for star-shaped and cyclical symmetries <cit.><cit.>, as well as more general interconnection topologies <cit.>-vidyasagar1979newmoylan1979tests<cit.>. However, the passivity verification for the networked system is centralized in these approaches, requiring information about the passivity of all subsystems. In contrast, we propose distributed dissipativity analysis conditions at the subsystem-level, whose feasibility is sufficient to guarantee dissipativity of the networked system. The subsystem-level conditions use only the knowledge of the subsystem dynamics and information about the dissipativity of its neighbors. Further, we do not impose any conditions on the network topology or homogeneity of the subsystem dynamics. 
    

    
  * Distributed synthesis: Using the distributed dissipativity analysis conditions, we then formulate a distributed procedure to synthesize local controllers
     at the subsystem-level to guarantee dissipativity of the networked system. The control synthesis is distributed in the sense that subsystems only use information about their dynamics and the dissipavity of the subsystems to which they are dynamically coupled, to design local subsystem-level controllers.
      

    
  * Compositionality: Finally, we propose an approach to design local controllers for networked systems which may be expanded by adding subsystems at a later stage. When a new subsystem is connected to the networked dynamical system, we formulate a control synthesis procedure that is compositional, that is, the design procedure uses only the knowledge of the dynamics of the newly added subsystem, and the dissipativity of its neighboring subsystems in the existing network, to synthesize local control inputs for the new subsystem, such that the new networked system is dissipative. This procedure does not require redesigning the existing controllers in the network when a new subsystem is added. 
     


In addition, we describe how the proposed synthesis approach can be extended to networks of switched systems. Such networks are encountered in several practical applications such as microgrids, where the dynamics and coupling between subsystems change with the availability of renewable generators. Therefore, extending our approach to this setting expands the applicability of our results to a larger class of applications. 
We illustrate the step-by-step implementation of the proposed distributed synthesis through a numerical example, and provide a case study demonstrating the application of this technique in enabling plug-and-play of generators in microgrids.

In <cit.>, we proposed a preliminary version of this approach to guarantee passivity for a limited class of networked systems with a cascade interconnection topology.  In this paper, we consider arbitrary network topologies, as well as a more general quadratic dissipativity framework, which allows us to capture a variety of properties of interest, such as, ℒ_2 stability, sector-boundedness, conicity, as well as passivity and its variants. We further extend the approach to networks of switched systems, which were not considered in <cit.>.

This paper is organized as follows. In Section <ref>, we describe the model of a networked system comprised of dynamically coupled subsystems. We then define dissipativity and formulate the problem of distributed synthesis of local controllers for this system in Section <ref>. In Section <ref>, we present results on the distributed verification of dissipativity, and distributed synthesis of local controllers to guarantee dissipativity of the networked system. In Section <ref>, we present a step-by-step illustration of the proposed  synthesis approach on a numerical example. We then extend these synthesis results to switched systems in Section <ref>, and demonstrate an application to microgrids in Section <ref>. The proofs of all the results in this paper are collected in the Appendix. 



Notation: We denote the sets of real numbers, positive real numbers including zero, and n-dimensional real vectors by ℝ, ℝ^+ and ℝ^n respectively. Define ℕ_N={1,…,N}, where N is a natural number excluding zero. Given a block matrix A=[A_i,j]_i∈ℕ_n,j∈ℕ_m, A_i,j represents the (i,j)-th block, and A'∈ℝ^n × m represents its transpose. Given matrices A_1,…, A_i, (A_1,…,A_i) represents a block-diagonal matrix with A_1,…, A_i as its diagonal entries. A symmetric positive definite matrix P ∈ℝ^n × n is represented as P>0 (and as P≥ 0, if it is positive semi-definite). The standard identity matrix is denoted by 𝐈, with dimensions clear from the context. Given sets A and B, A\ B represents the set of all elements of A that are not in B. 



§ SYSTEM DYNAMICS


Consider a networked dynamical system 𝐓_𝐍 comprised of N subsystems, as shown in Fig. 1, where the dynamics of the i-th subsystem Σ_i, i ∈ℕ_N is described by

    ẋ_i(t)     = A_i x_i(t) + B_i^(1)v_i(t) + B_i^(2) w_i(t) + B_i^(3) u_i(t), 
    
    y_i(t)     = C_ix_i(t), 
    
    v_i(t)     = ∑_j ∈ℕ_N H_i,j x_j(t),


where x_i(t) ∈ℝ^n_i, y_i(t) ∈ℝ^m_i,   v_i(t)∈ℝ^z_i, w_i(t)∈ℝ^l_i and u_i(t)∈ℝ^p_i are the state, output, coupling input, exogenous disturbance, and control input respectively. 

A subsystem Σ_j, j ∈ℕ_N\{i} is said to be dynamically coupled with the subsystem Σ_i if H_i,j≠ 0. Define the neighbor set for Σ_i to be 
    ℰ_i={j: H_i,j≠ 0, j ∈ℕ_N\{i}}.


The dynamics of the networked system 𝐓_𝐍 is written as
   
    ẋ(t)     = Ax(t) + B^(1)v(t) + B^(2)w(t) + B^(3)u(t)
    
    	y(t)     = Cx(t)
    
    	v(t)     = Hx(t)

where 
	
    A    =(A_1, A_2, …, A_N)
    
    	B^(j)   =(B^(j)_1, B^(j)_2,…,B^(j)_N),    j∈ℕ_3,
    
    	C    =(C_1, C_2, …, C_N) 
    
    	H     =  [ H_i,j ]_i,j ∈ℕ_N,

	and x(t) ∈ℝ^n, y(t) ∈ℝ^m,   v(t)∈ℝ^z, u(t)∈ℝ^p, and w(t)∈ℝ^l are the augmented system state, output, coupling input, control input, and disturbance formed by stacking x_i(t), y_i(t), v_i(t), u_i(t), and w_i(t) respectively of all N subsystems.  The matrix H represents the dynamical coupling in the system, and is referred to as the coupling matrix. 

We denote the interconnected system 𝐓_𝐍+1 obtained by connecting a new subsystem Σ_N+1 to 𝐓_𝐍 by 𝐓_𝐍+1:=𝐓_𝐍|Σ_N+1. 




§ PROBLEM DESCRIPTION

In this section, we formulate the problem of synthesizing local controllers at the subsystem-level in a distributed manner to enforce a quadratic dissipativity property on the interconnected system. 
 

<cit.>
 A dynamical system (<ref>) is said to be QSR-dissipative from w to y, if there exists a positive definite function V(x):ℝ^n⟶ℝ^+, called the storage function, such that, for all t>t_0≥0, x(t_0)∈ℝ^n, and w(t)∈ℝ^l,
 
 
    ∫_t_0^t[ y(τ); w(τ) ]'[  Q  S; S'  R ][ y(τ); w(τ) ] dτ≥ V(x(t))-V(x(t_0))

 

 
 holds, where x(t) is the state at time t resulting from the initial condition x(t_0), and Q, S and R are matrices of appropriate dimension. 
 

 We consider the objective of enforcing QSR-dissipativity on the networked system, as it can be used to capture a wide variety of dynamical properties of interest through appropriate choices of the Q, S and R matrices as follows. 
 

 <cit.>
System (<ref>) satisfying Definition <ref> is,

    
  * passive, if Q = 0, S = 1/2I and R =0,
    
  * strictly passive, if Q=-ρ I, S= 1/2I and R =-ν I, where ρ,ν∈ℝ^+\{0},
    
  * ℒ_2 stable, if Q=-1/γ I, S= 0 and R =γ I where γ∈ℝ^+ is an ℒ_2 gain of the system, 
    
  * conic, if Q=-I, S = c I and R = (r^2-c^2)I, where c∈ℝ and r ∈ℝ^+\{0}, and,
    
  * sector-bounded, if Q=-I, S = a+b/I and R = -ab I, where a,b∈ℝ.


 

Note that the QSR-dissipativity condition (<ref>) is a special case of an integral quadratic constraint (IQC) on (w,y), where the multiplier is an identity
matrix <cit.>. Under mild assumptions, the QSR-dissipativity property can also be used to guarantee Lyapunov stability <cit.>. QSR-dissipativity of the dynamical system (<ref>) can be analyzed as follows.
 

<cit.>
A dynamical system (<ref>) is QSR-dissipative with V(x)=x'Px if there exists a positive definite matrix P>0 and matrices Q, S and R of appropriate dimensions such that


    Γ = [ -Â'P-PÂ'+C'QC   -PB^(2)+C'S;  -B^(2)'P+S'C             R ]≥ 0


holds, where Â = A + B^(1)H.

 


The analysis condition in Proposition <ref> is centralized in the sense that a solution to (<ref>) requires knowledge of the dynamics and coupling of all subsystems in the networked dynamical system 𝐓_𝐍. However, for large-scale networks where new subsystems may be added or removed, it is desirable to develop an analysis and control synthesis that can be carried out locally at the subsystem-level with limited knowledge of the dynamics and coupling with neighboring subsystems. In this context, the aim of this paper is to address the following problems:

	 

    
  * Distributed analysis: Decompose the analysis condition in Proposition <ref> into conditions on the dissipativity of subsystems Σ_i, i ∈ℕ_N.
     

    
  * Distributed synthesis: Formulate a procedure to design local control inputs 
      

     
    u_i(t)   =∑_j ∈ℰ_i ∪{i}u_i,j(t),     i ∈ℕ_N,
    
        	u_i,j(t)   = K_i,jx_j(t),      j ∈ℰ_i ∪{i},

      

      
  such that 𝐓_𝐍 is QSR-dissipative, where the synthesis of the controller matrices K_i,j only uses the dynamics (<ref>), and information about the dissipativity of its neighboring subsystems Σ_j, j ∈ℰ_i.
     

    
  * Compositionality: When a new subsystem Σ_N+1 is connected to the networked dynamical system 𝐓_𝐍, obtain a control synthesis procedure which is compositional, that is, the design procedure uses only the knowledge of the dynamics of Σ_N+1, and the dissipativity of its neighboring subsystems Σ_j, j ∈ℰ_N+1, to synthesize local control inputs u_j,N+1(t), j∈ℰ_N+1, and 
    
    
    u_N+1(t)=∑_j ∈ℰ_N+1∪{N+1} u_N+1,j(t),
 
    
    such that 𝐓_𝐍+1:=𝐓_𝐍|Σ_N+1 is QSR-dissipative. 
     







§ DISTRIBUTED SYNTHESIS OF LOCAL CONTROLLERS

In this section, we present a distributed approach to synthesize local (subsystem-level) controllers that guarantee dissipativity of the networked dynamical system 𝐓_𝐍.  In this approach, every subsystem synthesizes a local controller using only the knowledge of its own dynamics, and information about the dissipativity of the subsystems to which it is dynamically coupled. The proofs of all the results in this section are collected in the Appendix.






 §.§ Distributed Analysis

We begin by distributing the dissipativity analysis condition in Proposition <ref>. We derive a property of positive definite matrices that will be useful in this context. 
  


A symmetric block matrix


    W = [ W_1,1 W_1,2     … W_1,N; W_2,1 W_2,2     … W_2,N;     ⋮     ⋮           ⋮; W_N,1 W_N,2     … W_N,N;       ],


where W_i,j, i,j ∈ℕ_N are block matrices of appropriate dimension, is positive definite if and only if

    
    M_i     > 0,  ∀ i ∈ℕ_N,  
    M_i     = W_i,i,     i =1  
      W_i,i - ∑_k=1^i-1 W_i,kM_k^-1 W_k,i,     i ∈ℕ\{1}.

    




The condition (<ref>) allows for the verification of the positive definiteness of a matrix to be carried out row-wise. Now, observe that the dissipativity of a networked dynamical system 𝐓_𝐍 can be analyzed by ascertaining the positive definiteness of matrix Γ in (<ref>). We can then use Lemma <ref> to decompose (<ref>) into conditions that can be verified at the subsystem-level. We have the following result.
 

 
The networked system 𝐓_𝐍 (<ref>) is QSR-dissipative from w to y with


    Q    = (Q_1, Q_2, …, Q_N),    Q_i ∈ ℝ^m_i × m_i
    
    S    = (S_1, S_2, …, S_N),    S_i ∈ℝ^m_i × l_i
    
    R    = (R_1, R_2, …, R_N),    R_i∈ℝ^l_i × l_i,


i ∈ℕ_N, if there exist matrices P_i,  termed energy matrices, such that



    ℙ_1:         P_i 
    
               P_i>0,  ℳ_i  > 0, 
        P_i ∈ℝ^n_i × n_i


is feasible ∀ i∈ℕ_N, where ℳ_i is computed in (<ref>).



Theorem <ref> provides distributed subsystem-level conditions for the verification of network-level dissipativity. The centralized analysis condition (<ref>) is decomposed into local conditions (<ref>), where each subsystem computes and stores an information matrix called as the messenger matrix ℳ_i, i ∈ℕ_N. The messenger matrix for each subsystem Σ_i, i ∈ℕ_N, is the difference between two terms, (i) μ_i^s, which can be interpreted as the dissipativity of the subsystem, and (ii) μ_i^c, which can be interpreted as the energy flow from its neighbors. The term μ_i^c contains information about the dynamical coupling between the subsystem Σ_i and its neighbors, as well as aggregated information about the dissipativity of its neighboring subsystems through the messenger matrices ℳ_j, and and energy matrices P_j, j ∈ℰ_i ∩ℕ_i-1, which are communicated by neighbors Σ_j, j ∈ℰ_i ∩ℕ_i-1. The positive definiteness of all messenger matrices (which can be verified at the subsystem-level) is sufficient to guarantee the dissipativity of the networked system 𝐓_𝐍. 



The analysis in Theorem <ref> can be implemented as described in Algorithm <ref>.



 We make the following remarks about the computation of the messenger matrix used in Theorem <ref>.

    
  (i) The computation of messenger matrix ℳ_i in (<ref>) requires ℳ_j^-1, j ∈ℰ_i∩ℕ_i-1. However, in cases where R_j=0, ℳ_j takes the form
	ℳ_j=  [ [             a_1,j             a_2,j; [2pt/2pt]
	a'_2,j                 0 ] ].
	Then, ℳ_i can be computed by replacing the expression for ℳ_j^-1 in (<ref>) by
	ℳ_j^-1=  [ [     a_1,j^-1            0; [2pt/2pt]
	0            0 ] ],
	 and relaxing the condition ℳ_j>0 to ℳ_j≥ 0 in ℙ_1. Note that this holds for all the results to follow, which will involve computation of messenger matrix.
 
  (ii) If the dynamics of subsystem Σ_i includes a feedthrough term D_i ∈ℝ^m_i× l_i such that y_i(t)=C_ix_i(t)+D_iw_i(t), then the computation of the messenger matrix ℳ_i in (<ref>) can be modified as follows. The term R_i in (<ref>)-(c) is modified to R_i ↦R̂_i, where R̂_i= -D_i'Q_iD_i-(D_i'S_i+S_i'D_i). 





In the i-th iteration of Algorithm <ref> (Steps 3-11), the dissipativity of the network 𝐓_𝐢 formed by the interconnection of subsystems Σ_1,Σ_2,⋯,Σ_i is verified. Therefore, the messenger matrices ℳ_i, i ∈ℕ_N will vary with the choice of numbering assigned to subsystems in the network, and the distributed analysis conditions in Theorem <ref> are only sufficient to guarantee dissipativity of the networked system 𝐓_𝐍.



In practice, the system matrices A_i, B_i, and C_i are known upto a certain level of modeling uncertainty. Since the proposed distributed analysis conditions only require the messenger matrix ℳ_i to be positive definite, it is possible to include a non-zero lower bound on ℳ_i which will account for uncertainties in the system model. For example, assume A_i has an additive uncertainty Δ A_i, that is, A_i ↦ A_i+Δ A_i. Then, the robustness margin for ℳ_i can be derived as follows. The term μ^s_i in (<ref>)-(c) is updated to μ̂^s_i, where μ̂^s_i = μ^s_i +  [ [ -(Δ A'_i P_i+P_i Δ A_i)                       0;            [2pt/2pt]
	0                       0 ] ]. The messenger matrix in Theorem <ref> is now updated to 
    ℳ̂_i = ℳ_i + [ [ -(Δ A'_i P_i+P_i Δ A_i)                       0;            [2pt/2pt]
    	0                       0 ]],

	and the verification condition ℳ_i>0 to ℳ̂_i>0, or 
	
    ℳ_i > [ [ (Δ A'_i P_i+P_i Δ A_i)                      0;           [2pt/2pt]
    	0                      0 ]].

	If Δ A_i is norm bounded, that is,  Δ A_i < ϵ_i then updating the verification condition ℳ_i>0 in Theorem <ref> to ℳ_i > ℳ_i^bound = [ [     2ϵ_i P_i            0; [2pt/2pt]
	0            0 ]] will ensure (<ref>), and hence the robustness of the proposed distributed analysis conditions to the additive uncertainty. The derivation of ℳ_i^bound for other classes of uncertainties like multiplicative or parametric uncertainties will require extending the proposed framework to a general integral quadratic constraint (IQC)  setup <cit.> , and is a subject of future work.




 §.§ Distributed Synthesis

Theorem <ref> provides sufficient conditions at the subsystem-level to guarantee dissipativity of the networked dynamical system 𝐓_𝐍. If the dissipativity conditions in Theorem <ref> are not met, we would like to synthesize local controllers at the subsystem-level to guarantee dissipativity of the networked system. Further, we require that the control synthesis be carried out at the subsystem-level, using only the dynamics of the subsystem and the messenger matrices communicated from its neighbors. We have the following result on distributed synthesis.



 The local control inputs 
 
 
    u_i(t)=∑_j ∈ℰ_i ∪{i}u_i,j(t),   i ∈ℕ_N,
       u_i,j(t)= K_i,jx_j(t),   j ∈ℰ_i ∪{i},

 
designed by solving



    ℙ_2:         P_i, K_i,i, K_i,j, K_j,i,    j ∈ℰ_i 
    
               P_i>0, 
       ℳ_i  > 0, 
    
        
        
        
        
        
        
        
              P_i ∈ℝ^n_i × n_i, 
        K_i,i∈ℝ^p_i × n_i, 
        K_i,j∈ℝ^p_i × n_j,   K_j,i∈ℝ^p_j × n_i,




for all i∈ℕ_N, where ℳ_i is the closed-loop messenger matrix of Σ_i computed from (<ref>), render 𝐓_𝐍 QSR-dissipative with



    Q    = (Q_1, Q_2, …, Q_N),    Q_i ∈ ℝ^m_i × m_i
    
    S    = (S_1, S_2, …, S_N),    S_i ∈ℝ^m_i × l_i
    
    R    = (R_1, R_2, …, R_N),    R_i∈ℝ^l_i × l_i,    i ∈ℕ_N.



The synthesis in Theorem <ref> can be carried out as described in Algorithm <ref>. 






At the i-th iteration of Algorithm <ref> (Steps 3-12), two types of controller matrices are designed at subsystem Σ_i to guarantee the dissipativity of the subnetwork 𝐓_𝐢 formed by the interconnection of subsystems Σ_1,Σ_2,⋯,Σ_i - (i) self controller matrix K_i,i, and (ii) coupling controller matrices K_i,j and K_j,i, j∈ℰ_i∩ℕ_i-1, corresponding to the bidirectional interconnections with its neighbors in 𝐓_𝐢. Note that the existing controllers in the subnetwork 𝐓_𝐢-1 are not redesigned at the i-th iteration, since the control synthesis at Σ_i is carried out to ensure the dissipativity of 𝐓_𝐢=𝐓_𝐢-1|Σ_i. 



	The energy flow between subsystems, μ_i^c, i ∈ℕ_N  in (<ref>), need not be small; therefore, the controller does not require or enforce weak coupling between subsystems.




The control synthesis equations in ℙ_2 are bilinear; however, they can readily be expressed as linear matrix inequalities using a Schur's complement method <cit.>. We also note that akin to any distributed approach, our synthesis yields more conservative controllers than those obtained by a centralized synthesis. 



Theorem <ref> relies on the distributed verification of dissipativity presented in Theorem <ref> to locally design linear subsystem-level controllers that guarantee the dissipativity of the networked system. While the underlying verification problem relies on the sufficient-only conditions in Theorem <ref> that may not always be feasible, the only consideration for the feasibility of the synthesis problem in Theorem <ref> is the existence of a controller that ensures the positive-definiteness of the messenger matrix in (<ref>). If all the subsystems in the network are minimum phase and R_i≥ 0 (or R̂_i≥0 for systems with feedthrough terms, where R̂_i is defined in Remark <ref>), then the linear feedback control design problem in Algorithm <ref> is always feasible, irrespective of the sequence (numbering) of subsystems used to solve the synthesis problem.  However, since (<ref>) is an LMI that can have multiple solutions, a secondary control objective may be used to choose a controller for a specific application. In this context, it is also possible to incorporate subsystem-level performance objectives in the distributed control synthesis problem while providing global dissipativity guarantees, since the addition of local optimization objectives at the subsystem-level does not affect the feasibility of the synthesis. However, the addition of control constraints may affect the feasibility of the synthesis problem by restricting the space of controllers for which the positive-definiteness of the messenger matrix can be guaranteed.







 §.§ Compositional Analysis and Control Synthesis

In large-scale networks, new subsystems may be connected to the existing network at a later time. In such scenarios, it is desirable to guarantee dissipativity of the updated network compositionally, that is, without redesigning pre-existing controllers. In this subsection, we extend the synthesis in Theorem <ref> to design local controllers for the newly added subsystem, using only limited information from its neighbors to guarantee dissipativity of the new networked system.







Given a QSR-dissipative system 𝐓_𝐍 with energy matrices P_i and messenger matrices ℳ_i, i ∈ℕ_N satisfying (<ref>) and (<ref>), or (<ref>) and (<ref>), and a new subsystem Σ_N+1, the new networked system  𝐓_𝐍+1:=𝐓_𝐍|Σ_N+1 is QSR-dissipative with new

    Q    ↦(Q,Q_N+1),    Q_N+1∈ℝ^m_N+1× m_N+1
    
     S    ↦(S,S_N+1),    S_N+1∈ℝ^m_N+1× l_N+1
    
     R    ↦(R,R_N+1),    R_N+1∈ℝ^l_N+1× l_N+1,

if there exist local control inputs 
 
    u_N+1(t)=∑_j ∈ℰ_N+1∪{N+1}u_N+1,j(t),
       u_N+1,j(t)= K_N+1,jx_j(t),  
       u_j(t)→ u_j(t)+u_j,N+1(t),
       u_j,N+1(t) = K_j,N+1x_N+1(t),

j ∈ℰ_N+1∪{N+1}, such that 

    ℙ_3:         P_N+1, K_N+1,j, K_j,N+1, K_N+1,N+1  j ∈ℰ_N+1
          P_N+1>0,
       ℳ_N+1  > 0, 
        P_N+1∈ℝ^n_N+1× n_N+1

is feasible, where ℳ_N+1 is computed in (<ref>).
 

Corollary <ref> can be implemented algorithmically according to Algorithm <ref>.

 



In <cit.>, a similar dissipativity-based approach is used to derive LMIs for the synthesis of distributed controllers, wherein the coupling of the LMIs (and, thereby, the underlying controllers) follows the same structure as the interconnection graph of the network. These LMIs can be distributed numerically and solved iteratively at each subsystem using the method of alternating projections or subgradient methods <cit.><cit.>. In contrast, the synthesis LMIs in (<ref>) are distributed and are solved only once at each subsystem. Furthermore, we pose our synthesis as a sequential procedure, which allows for compositional synthesis for networks that may be expanded by adding new subsystems, as described in Section <ref>-C. When a new subsystem is added, the control synthesis problem ℙ_3 is only solved at the new subsystem, and is independent of the size of the existing networked system.



§ NUMERICAL EXAMPLE

 

In this section, we present a numerical example to illustrate the distributed synthesis of local controllers for networked dynamical systems, and demonstrate the compositionality of the approach when new subsystems are added to the existing network. We begin by considering a
networked system 𝐓_3, comprised of three subsystems with dynamics and coupling given by



    Σ_1:    ẋ_1(t)     = [ -9  1;  5  7 ] x_1 (t) +  v_1(t) +[   1; 0.5 ] w_1(t) + u_1(t) 
          y_1(t)     = [ 3 2 ] x_1(t) 
          v_1(t)     = [  0.5 -0.7 ] x_1(t) + 0.1 x_2(t)


    Σ_2:    ẋ_2(t)     = 3 x_2 (t) + v_2(t) +  w_2(t) + u_2(t) 
          y_2(t)     = x_2(t) 
          v_2(t)     = [    1 -0.5 ] x_1(t) +  0.5 x_2(t) -0.1 x_3(t) .
       Σ_3:    ẋ_3(t)     = - x_3 (t) +  v_3(t) +  w_3(t) +  u_3(t) 
          y_3(t)     = x_3(t) 
          v_3(t)     = -0.7 x_2(t) + 0.2 x_3(t) .


The objective is to guarantee passivity of the networked system according to the definition in Remark <ref>-(1). 

We begin by checking if Σ_1 is passive using Algorithm <ref>, and compute  controller matrix K_1,1 to guarantee passivity of Σ_1. We also compute the closed loop messenger and energy matrices ℳ_1 and P_1 respectively of Σ_1. We then use ℳ_1 and P_1 communicated from Σ_1, and the dynamics of Σ_2 to 



verify the sufficient conditions in Theorem <ref> for the network comprised of Σ_1 and Σ_2. Since the sufficient conditions are not satisfied, we use the procedure in Algorithm <ref> to synthesize controller matrices K_2,2, K_2,1 and K_1,2 at Σ_2 to guarantee passivity of the interconnection of Σ_1 and Σ_2. Additionally, we compute messenger matrix ℳ_2 and energy matrix P_2 at Σ_2. Next, we use the dynamics of the subsystem Σ_3, and ℳ_2 and P_2 communicated from Σ_2 to Σ_3 in Algorithm <ref>. Since ℙ_1 is feasible (Step 6 in Algorithm <ref>), the networked system 𝐓_3 comprised of the interconnection of Σ_3 with Σ_1 and Σ_2 is passive, and no controller design is required at Σ_3. 

The controller matrices K_3,3, K_2,3 and K_3,2 are set to zero, and messenger matrix ℳ_3 and energy matrix P_3 are computed and stored at Σ_3. The networked system 𝐓_3 with its control architecture is shown in Fig. <ref>, and the controller matrices are as shown in Fig. <ref>.  


Now consider the networked system 𝐓_4=𝐓_3|Σ_4 formed by adding a new subsystem Σ_4 to 𝐓_3 as shown in Fig. <ref>(a), where Σ_4 is dynamically coupled to Σ_1 and Σ_2. The dynamics of Σ_4 is given by


    Σ_4: ẋ_4(t)     =[   2   1;   3 0.8 ] x_4(t) +[ 1.2; 0.8 ] v_1(t) +[  0.5; -0.2 ] w_4(t)
        +[ 1.2; 0.8 ] u_4(t) 
    
    y_4(t)     =[ 2.1 0.6 ] x_4(t) 
    
    v_4(t)     =[ -0.9 -0.3 ] x_1(t) - 0.9 x_2(t) +[ 1.1 0.4 ] x_4(t).
 


Additionally, the coupling inputs v_1(t) and v_2(t) are updated to,

    v_1(t)    = [  0.5 -0.7 ] x_1(t) + 0.1 x_2(t)+[ 0.2 0.2 ]x_4(t), 
    
    v_2(t)     = [    1 -0.5 ]x_1(t) +  0.5 x_2(t) -0.1 x_3(t)+[ 0.2 0.2 ]x_4(t).

At subsystem Σ_4, we use matrices ℳ_1, P_1, ℳ_2 and P_2 received from Σ_1 and Σ_2 (its neighboring subsystems) in the compositional synthesis procedure described in Algorithm <ref> to design controller matrices K_4,4, K_1,4, K_4,1, K_4,2 and K_2,4 that guarantee passivity of the networked system 𝐓_4=𝐓_3|Σ_4. The compositional control synthesis procedure is illustrated in Fig. <ref>. The distributed synthesis algorithm allows for dynamics of subsystems to be dissimilar and of different dimensions, as long as the network is `proper', that is, the input-output dimensions are suitable to define the interconnections between subsystems.




We would like to highlight the fact that the proposed algorithm distributes the analysis and control synthesis between subsystems  sequentially. As described in Remark <ref>, Algorithm <ref> is generally always feasible (with only one exception). In the next part of this example, we demonstrate the feasibility of the control synthesis by reapplying the distributed synthesis approach to the same network, but solving the synthesis problem in Algorithm <ref> in two different sequences as shown in Fig. <ref> and Fig. <ref>.


  


  §.§.§ Solution sequence {3-2-1-4}

Consider the set-up shown in Fig. <ref>. As opposed to the discussion so far in this example, where the distributed synthesis Algorithm <ref> solves ℙ_1 (and ℙ_2) for Σ_1 first, followed by Σ_2, Σ_3, and Σ_4, in that order, we now
begin by analyzing Σ_3 first, followed by Σ_2, Σ_1, and Σ_4, in that order. The design steps are summarized below:

    
  (a) We use the sufficient conditions in Theorem <ref> to verify that Σ_3 is passive. Therefore, no controller design is required at Σ_3, and we set K_3,3=0. We also compute and store the messenger and energy matrices ℳ_3 and P_3 respectively of Σ_3. 
    
  (b) We then consider the network 𝐓_2, comprised of the interconnection of Σ_3 and Σ_2, as shown in Fig. <ref>-(b). We use the messenger and energy matrices ℳ_3 and P_3 respectively, communicated from Σ_3, along with the dynamics of Σ_2 in Algorithm <ref> to verify the passivity of this network. Since the passivity conditions for 𝐓_2 are not met, we use Step 10 of Algorithm <ref> to synthesize controller matrices K_2,2, K_2,3, and K_3,2 at Σ_2 to guarantee passivity of interconnection of Σ_3 and Σ_2. In this case, the controller matrices K_2,3, and K_3,2 turn out to be zero. We also compute messenger matrix ℳ_2 and energy matrix P_2 at Σ_2.
    
  (c) Next, we consider the network 𝐓_3=𝐓_2|Σ_1, as shown in Fig. <ref>-(c). We use the dynamics of the subsystem Σ_1, and ℳ_2 and P_2 communicated from Σ_2 to Σ_1 in Algorithm <ref> to verify that the network 𝐓_3 is not passive, that is, problem ℙ_1 is infeasible. Therefore, we solve ℙ_2 to design controller matrices 
    K_1,1, K_1,2, and K_2,1 at Σ_1 to guarantee passivity of the networked system 𝐓_3 comprised of the interconnection of Σ_3, Σ_1 and Σ_2. We also compute messenger matrix ℳ_1 and energy matrix P_1 at Σ_1.
    
  (d) Lastly, we apply Algorithm <ref> to subsystem Σ_4. The dynamics of the subsystem Σ_4, and messenger matrices ℳ_1 and ℳ_2, and energy matrices P_1 and P_2, communicated to Σ_4 from its neighboring subsystems Σ_1 and Σ_2, are used in ℙ_1 and ℙ_2 to solve for the controller gains K_4,4, K_4,1, K_4,2, K_1,4, and K_2,4. The closed loop system 𝐓_4 is guaranteed to be passive.  


The designed controller matrices are shown in Fig. <ref>. Note that although the distributed synthesis algorithm was applied in the sequence {3-2-1-4}, the synthesis problem is feasible, and the designed local controllers look very similar to the ones designed when the synthesis algorithm was solved in the sequence {1-2-3-4}.


  §.§.§ Solution sequence {3-4-1-2}

A more interesting scenario is represented in Fig. <ref>, where the sequence in which Algorithm <ref> is applied starts from Σ_3, and moves on to Σ_4 in the next step, to guarantee passivity of the network 𝐓_2 comprised of Σ_3 and Σ_4.  Since the two subsystems, Σ_3 and Σ_4, have no direct coupling, 𝐓_2 is not fully connected. 


However, the results presented in Theorem <ref> and Theorem <ref> do not impose any restrictions on the topology of the network and can be applied regardless of the connectivity of the graph. Therefore, when the distributed synthesis algorithm is applied to Σ_4 here, it uses the dynamics of Σ_4 in ℙ_1 and ℙ_2 to compute controller gain K_4,4, and messenger matrix ℳ_4 and energy matrix P_4. The messenger and energy matrices of Σ_3 are not communicated to Σ_4 because Σ_3 and Σ_4 are not coupled. In the next step, ℳ_4 and P_4 are communicated to Σ_1 from Σ_4 and Algorithm <ref> is applied to Σ_1. Note that at this stage, the network 𝐓_3 comprised of Σ_1, Σ_3, and Σ_4 is still not fully connected. However, the proposed distributed synthesis approach still works to guarantee the passivity of the network 𝐓_3 until this point. Similarly, Σ_2 is added to the network and Algorithm <ref> is applied to Σ_2, using its dynamics, and the messenger and energy matrices, ℳ_1, ℳ_3, ℳ_4, 𝒫_1, 𝒫_3 and 𝒫_4, received from its neighbors Σ_1, Σ_3 and Σ_4, to guarantee passivity of the networked system 𝐓_4. The steps involved in the synthesis are illustrated in Fig. <ref>, and the designed controller matrices are shown in Fig. <ref>.


Note that the controller gains designed at Σ_2 using this solution sequence are quite different from the ones designed in Fig. <ref> and Fig. <ref>. This difference can be interpreted in the context of Remark <ref> as follows. For the solution sequence {1-2-3-4}, when the control design process is

carried out at Σ_2, the term μ_2^c in the messenger matrix ℳ_2 corresponding to the energy flow from its neighbors only comprises of the energy flow from subsystem Σ_1. Similarly, for the solution sequence {3-2-1-4}, the messenger matrix ℳ_2 only contains the energy flow from the neighbor Σ_3. However, when the control synthesis algorithm is solved in the sequence {3-4-1-2}, the subsystem Σ_2 is added in the last step of the procedure, and the term μ_2^c in the messenger matrix ℳ_2 comprises of energy flows from all its neighboring subsystems, Σ_1, Σ_2, and Σ_3. This results in large gains for the controllers designed at Σ_2 to ensure the positive definiteness of the messenger matrix ℳ_2. This example highlights the trade-off involved in the proposed distributed synthesis, introduced by the sequential nature of the synthesis algorithm. 

In achieving distributed synthesis, Algorithm <ref> introduces some conservatism, where local controllers may be designed at an intermediate step of the synthesis procedure, even if the networked system as a whole is dissipative. Intuitively, the design of redundant or unnecessary controllers will decrease with the number of subsystems into which the networked system is divided, with a purely centralized synthesis being the least conservative in this regard. However, we note that it is precisely this sequential nature of the synthesis that allows compositionality for networks that may be expanded by adding new subsystems, as described in Section <ref>-C.










§ EXTENSION TO SWITCHED SYSTEMS

In many emerging applications of large-scale networked systems in infrastructure networks, the subsystem dynamics, and even the coupling matrices between subsystems can change during operation. For example, in power grids comprised of interconnected microgrids, both the dynamics of individual microgrids and the coupling between microgrids change when a new microgrid is connected to the network, and with changes in the operating point <cit.>. In order to synthesize local controllers in a distributed manner and guarantee compositionality for such applications, we extend the distributed synthesis presented in Section <ref> to networks of switched systems, where subsystem dynamics is time-varying. 








Consider a networked system 𝐓_𝐍^σ comprised of N subsystems, where the dynamics of the i-th subsystem Σ_i is switching and is given by



    ẋ_i(t)     = A_i^σ_i(t) x_i(t) + B_i^(1)^σ_i(t)v_i(t) + B_i^(2)^σ_i(t) w_i(t) 
          + B_i^(3)^σ_i(t) u_i(t), 
    
    y_i(t)     = C_i^σ_i(t) x_i(t), 
    
    v_i(t)     = ∑_j ∈ℕ_N H_i,j x_j(t),


where x_i(t), y_i(t), v_i(t), u_i(t), and w_i(t) are as described in Section <ref>. The system matrices  A_i^σ_i(t), B_i^(1)^σ_i(t), B_i^(2)^σ_i(t), B_i^(3)^σ_i(t), and C_i^σ_i(t) vary based on the value of the switching signal σ_i(t): ℝ^+→ℕ_η_i, where η_i is the number of switching modes of Σ_i. Note that we do not place any restrictions on the sequence in which the dynamics of Σ_i switches, and do not require the switching sequence to be known a priori. 

Now, the dynamics of 𝐓_𝐍^σ is described by


    ẋ(t)     = A^σ(t)x(t) + B^(1)^σ(t)v(t) + B^(2)^σ(t)w(t) 
          + B^(3)^σ(t)u(t)
    
    y(t)     = C^σ(t)x(t)
    
    v(t)     = Hx(t)


where

    A^σ(t)   =(A_1^σ_1(t), A_2^σ_2(t), …, A_N^σ_N(t))
    
    B^(j)^σ(t)   = (B_1^(j)^σ_1(t), B_2^(j)^σ_2(t), …,B_N^(j)^σ_N(t)),    j∈ℕ_3
    
    C^σ(t)   =(C_1^σ_1(t), C_2^σ_2(t), …, C_N^σ_N(t))
    
    H     =  [ H_i,j ]_i,j ∈ℕ_N


and x(t), y(t), v(t), u(t), w(t) and σ(t) are the augmented system state, output, coupling input, control input, disturbance and switching signal formed by stacking x_i(t), y_i(t), v_i(t), u_i(t), w_i(t) and σ_i(t) respectively of all N subsystems. 

As described in <cit.>, the classical form of dissipativity in Definition <ref> holds for switched system (<ref>) as well. Along the lines of Section <ref>, we have the following result on distributed synthesis of local controllers to guarantee dissipativity of the networked switched system 𝐓_𝐍^σ. 


	The local control inputs 
	
	
    u_i(t)=∑_j ∈ℰ_i ∪{i}u_i,j(t),    i ∈ℕ_N,
       u_i,j(t)= K_i,jx_j(t),   j ∈ℰ_i ∪{i},

	
designed by solving 

	
    ℙ_4:         P_i, K_i,i, K_i,j, K_j,i  j ∈ℰ_i 
          P_i>0,
       ℳ_i  > 0, 
    
    	
    	
    	
    	
    	
    	
    	
    	    P_i ∈ℝ^n_i × n_i, 
       K_i,i∈ℝ^p_i × n_i, 
        K_i,j∈ℝ^p_i × n_j,   K_j,i∈ℝ^p_j × n_i,

	
for all i∈ℕ_N and all σ_i ∈ℕ_η_i, σ_j ∈ℕ_η_j, j ∈ℰ_i ∩ℕ_i-1, where ℳ_i is computed from (<ref>), render 𝐓_𝐍^σ (<ref>) QSR-dissipative with

    Q    = (Q_1, Q_2, …, Q_N),    Q_i ∈ ℝ^m_i × m_i
    
    S    = (S_1, S_2, …, S_N),    S_i ∈ℝ^m_i × l_i
    
    R    = (R_1, R_2, …, R_N),    R_i∈ℝ^l_i × l_i,    i ∈ℕ_N.

The closed-loop messenger matrix of Σ_i is then given by ℳ_i=min_σ_i,σ_j ||ℳ_i||.




The messenger matrix ℳ_i in the distributed synthesis result of Theorem <ref> corresponds to the least dissipative mode of Σ_i. This allows for a reduction in the computational complexity of the control synthesis arising from a possibly large number of switching modes in the networked system. If the coupling matrix H is also switching, then the maximum value of the coupling term μ_i,σ_i^c in (<ref>) over all possible switching sequences of Σ_j, j∈ℰ_i ∩ℕ_i-1 can be used in (<ref>).







 Note that Theorem <ref> is based on using the same energy matrix P_i for all switching modes of Σ_i in (<ref>). We can allow different energy matrices in different switching modes of the subsystem for specific switching signals that are known a priori. However, it can be shown that a dissipative switched system under arbitrary switching can not have different energy matrices in different switching modes (see Appendix for details).





The compositionality results, as well as the verification and synthesis algorithms in Section <ref> can similarly be extended to networks of switched systems. 


In <cit.>, it was shown that the passivity of a nonlinear switched system in a neighborhood around any operating point can be inferred from that of its linear approximation. These results were further extended in <cit.> to guarantee the dissipativity of a network of nonlinear switched systems through controllers designed in a centralized manner, using the network formed by linear approximations of the nonlinear switched systems. It is easy to see that similar linear approximation arguments can be used to extend the distributed synthesis procedure proposed in this paper to guarantee dissipativity of a networked system comprised of nonlinear subsystems. Additionally, recently developed notions of equilibrium-independent dissipativity (passivity) <cit.><cit.> can be explored in the context of guaranteeing dissipativity around non-trivial operating points. 




§ CASE STUDY: MICROGRID NETWORK

In this section, we consider the problem of compositional synthesis of local controllers for power networks with large-scale integration of renewables. In such networks, several small distributed generation units (DGUs) and loads are aggregated in clusters known as microgrids. In microgrids, since renewable inputs like wind speed and solar intensity vary continuously, and DGUs either participate or do not participate in the network depending on availability and requirement, switching dynamics are inherent <cit.>. Therefore, it is necessary to synthesize local controllers for DGUs in a compositional manner, such that the stability of the microgrid is maintained when new DGUs connect to the grid, without requiring redesign of existing DGU controllers. In this section, we demonstrate the application of our distributed synthesis framework to enable this `plug-and-play' operation of DGUs in a microgrid.



We consider a microgrid network with three DGUs as shown in Fig. <ref>. Each DGU is modeled as a voltage source with internal voltage V_ti, connected to an RLC-circuit with resistance, inductance and capacitance given by X_R,ti, X_L,ti and X_C,ti respectively. The internal voltage at the DGU is stepped up by a transformer with turn ratio k_i to obtain a terminal voltage V_i. The line connecting the i-th and j-th DGUs is assumed to have an impedance Z_ij=X_R,ij+√(-1)ω_0 X_L,ij, where X_R,ij and X_L,ij are the resistance and inductance of the line respectively, and ω_0 is the base frequency of the network. 

The dynamics of the microgrid can be modeled as a networked switched system, with system matrices given by (<ref>). The system parameters are provided in Table <ref> <cit.>. As shown in Fig. <ref>, DGU-3 can either connect or disconnect to DGU-1 in the network. The dynamics of the i-th DGU switches based on the set of DGUs 𝒮_i,σ_i, to which it is connected, as described in (<ref>). The states (and outputs) of each DGU comprise of the direct and quadrature axis components <cit.> of the terminal voltages (denoted by V_i,d and V_i,q respectively) and internal currents at the DGU unit (denoted by I_ti,d and I_ti,q respectively). The control inputs comprise of the direct and quadrature axis internal voltages of the DGU, denoted by V_ti,d and V_ti,q respectively. The disturbances correspond to the direct and quadrature axis line currents drawn from the DGU, denoted by I_Li,d and I_Li,q respectively, which vary based on fluctuations in the power sharing between DGUs.








Using Theorem <ref>, we design local controllers for all DGUs to guarantee ℒ_2 stability of the network, by choosing Q_i=-𝐈, S_i=0 and R_i=γ_i^2 𝐈, i ∈ℕ_3, where γ_i represents the ℒ_2 gain of the closed loop dynamics of Σ_i. The parameters γ_i are considered as variables in the synthesis problem ℙ_2, and are found to be

    γ_1   =2.85
    γ_2   =3.21
    γ_3   =3.22.


We consider a test scenario where DGU-3 connects to the network at t=1s and disconnects at t=3s, causing a transient in the system states. The terminal voltage profiles (states) at DGU-1 and DGU-3 during this operation are shown in Fig. <ref>, clearly demonstrating that the proposed controllers maintain the stability of the network during plug-and-play operation. 








§ CONCLUSION

We presented a distributed and compositional approach to synthesize local controllers for networked systems comprised of dynamically coupled subsystems. The proposed approach can readily be extended to guarantee local dissipativity properties for nonlinear networked systems operating close to equilibrium. Future work will involve extending the distributed synthesis approach to more general classes of nonlinear and hybrid systems.





Proof of Lemma <ref>
Consider lower triangular matrices L_i,i,  i∈ℕ_N and matrix 



    L   = [ L_i,j ]_i,j∈ℕ_N, 
    
    L_i,j   = 
    0      j>i
    
    W_i,jL_j,j'^-1    j ∈{2,…,N},  j<i
    
    
    L_i,iL'_i,i   =W_i,i    i =1
    W_i,i-∑_j∈ℕ_i-1 L_i,jL'_i,j    i ∈{2,…,N}


with W_i,j, i,j ∈ℕ_N being elements of W as defined in (<ref>). Define M_i = L_i,iL'_i,i, ∀ i∈ℕ_N. 

A symmetric matrix M_i is positive definite if and only if there exists a lower triangular matrix L_i,i with positive diagonal entries such that M_i = L_i,iL'_i,i <cit.>. Therefore, if (<ref>) holds, L_i,i will exist with positive diagonal entries. Invertibility of L_i,i,   i∈ℕ_N guarantees the existence of L_i,j,   i,j∈ℕ_N,   j<i. Thus, we can always find a lower triangular matrix L of the form (<ref>), with positive diagonal entries, such that W = LL'. This implies the positive definiteness of W <cit.>, proving the sufficiency of Lemma <ref>.  Along similar lines, we can also prove the necessity of (<ref>) for the positive definiteness of W.  ▪





Proof of Theorem <ref>


From Proposition <ref>, (<ref>) is QSR-dissipative if
 
  
    Γ =  [ [               -Â'P-PÂ+C'QC                -PB^(2)+C'S; [2pt/2pt]
        -B^(2)'P+S'C                          R ] ]≥ 0,


where Â = A + B^(1)H. Consider Q = (Q_1,Q_2,…, Q_N), S = (S_1,S_2,…, S_N), R = (R_1,R_2,…, R_N) and P = (P_1,P_2,…, P_N), where Q_i∈ℝ^m_i × m_i, S_i∈ℝ^m_i × l_i, R_i∈ℝ^l_i × l_i and P_i∈ℝ^n_i × n_i, i∈ℕ_N. Consider a permutation matrix
  
    =0pt
      𝔼 =  [[   e_1' e_N+1'   e_2' e_N+2'      ⋯   e_N' e_N+N' ]]',

where e_k,   k∈ℝ^+\{0}, are defined as e_i=[e_i^a,0_n_i,l], and e_N+i=[0_l_i,n,e_i^b], where 

    
  * e_i^a is a matrix with dimension n_i× n, that contains all zero elements, but an identity matrix of dimension n_i at columns (∑_j=1^i-1 n_j)+1:(∑_j=1^i n_j).
    
  * e_i^b is a matrix with dimension l_i× l, that contains all zero elements, but an identity matrix of dimension l_i at columns (∑_j=1^i-1 l_j)+1:(∑_j=1^i l_j).
    
  * 0_r,c is a matrix, with r rows and c columns, whose entries are all 0.

Right multiplication of Γ with 𝔼' permutes its columns, and a left multiplication with 𝔼 permutes its rows.
  
  
    𝔼Γ𝔼'     = W = [ W_1,1 W_1,2     … W_1,N; W_2,1 W_2,2     … W_2,N;     ⋮     ⋮           ⋮; W_N,1 W_N,2     … W_N,N;       ],
    
    
        W_i,i    = [ [   -Â_i'P_i-P_iÂ_i'
        +C_i'Q_iC_i                -P_iB_i^(2)+C_i'S_i; [2pt/2pt]
        -B_i^(2)'P_i+S_i'C_i                                R_i ]], 
     
        W_i,j    = 
        [[   -Ĥ_i,j-Ĥ'_j,i               0; [2pt/2pt]
        0               0 ]],

  
for all i,j∈ℕ_N,  j≠ i, where Â_i = A_i + B_i^(1)H_i and Ĥ_i,j = P_iB_i^(1)H_i,j. 

Note that Γ≥ 0 if and only if 𝔼Γ𝔼' ≥ 0. If (<ref>) and (<ref>) hold, then, from Lemma <ref>, 𝔼Γ𝔼'>0 and all conditions in Proposition <ref> are satisfied with Q = (Q_1,Q_2,…, Q_N), S = (S_1,S_2,…, S_N), R = (R_1,R_2,…, R_N) and P = (P_1,P_2,…, P_N). Therefore, the networked dynamical system 𝐓_𝐍 in (<ref>) is QSR-dissipative.  
▪   




Proof of Theorem <ref>


The proof follows by applying Theorem <ref> to the closed loop system,

    ẋ_i(t)     = A_i x_i(t) + B_i^(1)v_i(t) + B_i^(2) w_i(t) 
        + B_i^(3)∑_j ∈ℰ_iK_i,j(t)x_j(t), 
    
    y_i(t)     = C_ix_i(t), 
    
    v_i(t)     = ∑_j ∈ℕ_N H_i,j x_j(t).
▪





Proof of Corollary <ref>

If ℙ_3 is feasible, then Theorem <ref> holds for 𝐓_𝐍+1=𝐓_𝐍|Σ_N+1, thus completing the proof.
▪




Proof of Theorem <ref>

Since Definition <ref> holds for switched systems <cit.>, along the lines of proof for Theorem <ref>, the networked switched system (<ref>) with 
    u_i(t)=∑_j ∈ℰ_iK_i,jx_j(t),
 i∈ℕ_N, is QSR-dissipative if

 
    W    = [ W_1,1 W_1,2     … W_1,N; W_2,1 W_2,2     … W_2,N;     ⋮     ⋮           ⋮; W_N,1 W_N,2     … W_N,N;       ] > 0



holds, where

    W_i,i    = [ [               -Â_i^σ_i'P_i-P_iÂ_i^σ_i';                    +C_i^σ_i'Q_iC_i^σ_i            -P_iB_i^(2)^σ_i+C_i^σ_i'S_i; [2pt/2pt]
    -B_i^(2)^σ_i'P_i+S_i'C_i^σ_i                                    R_i ]],
    
    W_i,j    =
    [[ Ĥ_j,i^σ_j'+Ĥ_i,j^σ_i                    0;          [2pt/2pt]
    0                    0 ]],


Â_i^σ_i = A_i^σ_i + B_i^(1)^σ_iH_i,  Ĥ_i,j^σ_i = P_i(B_i^(1)^σ_iH_i,j+B_i^(3)^σ_iK_i,j) and i,j ∈ ℕ_N,  j≠ i and σ_i ∈ℕ_η_i. Clearly, if (<ref>) and (<ref>) hold, then, from Lemma <ref>, the closed loop networked switched system 𝐓_𝐍^σ is QSR-dissipative.
▪




Note on Remark <ref>


Consider a  switched system Σ

    ẋ(t)     = A^σ(t)x(t) + B^(1)^σ(t)v(t) + B^(2)^σ(t)w(t)
    + B^(3)^σ(t)u(t)
    
    y(t)     = C^σ(t)x(t)

which switches arbitrarily between two modes σ(t) ∈{1,2}. Suppose Σ is  QSR-dissipative with multiple energy matrices (or multiple storage functions), that is, if σ(t)=1, ∀ t_0≤ t ≤ t_1, Σ satisfies the dissipativity inequality
 
    ∫_t_0^t_1[ y(τ); w(τ) ]'[  Q  S; S'  R ][ y(τ); w(τ) ] dτ≥ V_1(x(t_1))-V_1(x(t_0)),

 where V_1(x)=x'P_1x, and if σ(t)=2, ∀ t_0≤ t ≤ t_1, Σ satisfies
 
    ∫_t_0^t_1[ y(τ); w(τ) ]'[  Q  S; S'  R ][ y(τ); w(τ) ] dτ≥ V_2(x(t_1))-V_2(x(t_0)),

 where V_2(x)=x'P_2x.
 
 If the dynamics of Σ switches from mode 1 (σ=1) to mode 2 (σ=2) at time t, then, σ(t^-)=1 and σ(t^+)=2. Then, 
 
    ∫_t^-^t^+[ y(τ); w(τ) ]'[  Q  S; S'  R ][ y(τ); w(τ) ] dτ≥ V_2(x(t))-V_1(x(t))

  must hold. Since Σ is dissipative for arbitrary switching, consider a different switching signal where Σ switches from  mode 2 (σ=2) to mode 1 (σ=1) at time t. Then,
   
    ∫_t^-^t^+[ y(τ); w(τ) ]'[  Q  S; S'  R ][ y(τ); w(τ) ] dτ≥ V_1(x(t))-V_2(x(t)).

  must hold. 
  
  Clearly, both (<ref>) and (<ref>) can hold if and only if V_1(x) = V_2(x), that is, the energy matrices P_1 and P_2 are the same. A similar argument follows for dynamical systems with more than two switching modes. It is therefore not possible to have different energy matrices in different modes for a switched system that is dissipative for arbitrary switching.


IEEEtran
 






[
    < g r a p h i c s >
]Etika Agarwal
obtained her PhD in Electrical Engineering from the University of Notre Dame in 2019. She received her Masters in Electrical Engineering from the University of Notre Dame in 2016, and her B. Tech in Avionics from the Indian Institute of Space Science and Technology in 2012. Before joining the graduate school, she worked with the Indian Space Research Organization from 2012-2014. Her research interests are in computationally efficient and scalable control of large-scale cyber-physical systems. 



[
    < g r a p h i c s >
]S Sivaranjani
obtained her PhD in Electrical Engineering from the University of Notre Dame in 2019. She obtained her undergraduate and Master’s degrees in Electrical Engineering from the PES Institute of Technology and the Indian Institute of Science, in 2011 and 2013, respectively. Her research interests are in the area of distributed control for large-scale infrastructure networks, with emphasis on transportation networks and power grids. She is a recipient of the Schlumberger Foundation Faculty for the Future fellowship (2015-2018), the Zonta International Amelia Earhart fellowship (2015-2016) and the Notre Dame (NSF) Ethical Leaders in STEM fellowship (2016-2017).






[
    < g r a p h i c s >
]Vijay Gupta
is a Professor in the Department of Electrical Engineering
	at the University of Notre Dame, having joined the faculty in January
	2008. He received his B. Tech degree at Indian Institute of
	Technology, Delhi, and his M.S. and Ph.D. at California Institute of
	Technology, all in Electrical Engineering. Prior to joining Notre
	Dame, he also served as a research associate in the Institute for
	Systems Research at the University of Maryland, College Park. He
	received the 2018 Antonio Ruberti Award from IEEE Control Systems
	Society, the 2013 Donald P. Eckman Award from the American Automatic
	Control Council and a 2009 National Science Foundation (NSF) CAREER
	Award. His research and teaching interests are broadly at the
	interface of communication, control, distributed computation, and
	human decision making.


[
    < g r a p h i c s >
]Panos Antsaklis
is the H.C. & E.A. Brosey Professor of Electrical Engineering at the University of Notre Dame. He is graduate of the National Technical University of Athens, Greece, and holds MS and PhD degrees from Brown University. His research addresses problems of control and automation and examines ways to design control systems that will exhibit high degree of autonomy. His current research focuses on Cyber-Physical Systems and the interdisciplinary research area of control, computing and communication networks, and on hybrid and discrete event dynamical systems. He is IEEE, IFAC and AAAS Fellow, President of the Mediterranean Control Association, the 2006 recipient of the Engineering Alumni Medal of Brown University and holds an Honorary Doctorate from the University of Lorraine in France. He served as the President of the IEEE Control Systems Society in 1997 and was the Editor-in-Chief of the IEEE Transactions on Automatic Control for 8 years, 2010-2017.





