Active Nodes of Network Systems With Sum-Type Dissipation Inequalities

This article studies a small gain-like analysis and control synthesis tool for large-scale interconnected dynamical systems (networks). By exploiting the structure of the interconnection one can derive (or enforce it via control synthesis) an algebraic condition (the small gain-like condition) to allow the construction of a network storage function with a network dissipation inequality in a desired form (the small gain-like property), which can be further used for establishing convergence or stability properties. Small gain-like conditions, for systems with quadratic, general-form nonlinear, and parametrized supply rates, respectively, are derived and interpreted using the underlying graph. This allows enforcing such a condition via the design of a class of controlled nodes, called active nodes, provided their locations satisfy a graph-based condition. The article then proceeds to discuss control synthesis methods using the notion of active nodes, including the placement, the parameter computation, and the adaptation of the active nodes. Finally, an example of public-health-related control for interconnected settlements, to demonstrate the implementation of the active-node-based scheme, is presented.


Active Nodes of Network Systems With
Sum-Type Dissipation Inequalities Kaiwen Chen , Member, IEEE, and Alessandro Astolfi , Fellow, IEEE Abstract-This article studies a small gain-like analysis and control synthesis tool for large-scale interconnected dynamical systems (networks).By exploiting the structure of the interconnection one can derive (or enforce it via control synthesis) an algebraic condition (the small gainlike condition) to allow the construction of a network storage function with a network dissipation inequality in a desired form (the small gain-like property), which can be further used for establishing convergence or stability properties.Small gain-like conditions, for systems with quadratic, general-form nonlinear, and parametrized supply rates, respectively, are derived and interpreted using the underlying graph.This allows enforcing such a condition via the design of a class of controlled nodes, called active nodes, provided their locations satisfy a graph-based condition.The article then proceeds to discuss control synthesis methods using the notion of active nodes, including the placement, the parameter computation, and the adaptation of the active nodes.Finally, an example of publichealth-related control for interconnected settlements, to demonstrate the implementation of the active-node-based scheme, is presented.Index Terms-Interconnected systems, large-scale systems, Lyapunov methods, networked control systems.

I. INTRODUCTION
D YNAMICAL systems interconnected in a network struc- ture have seen extensive research since the second half of last century, under the names of dynamical network systems and large-scale interconnected systems.In particular, stability analysis (see, e.g., [1], [2], [3], [4], and [5]) has been one of the main research areas since, as it is well known, even if each subsystem (node) possesses some stability properties when disconnected from the overall system (network), such properties are not necessarily preserved under interconnection.
The majority of the works that guarantee network stability properties via preservation of node stability properties (e.g., input-to-state stability (ISS) [6], integral input-to-state stability (iISS) [7], and input-to-output stability (IOS) [8]) under interconnection are based on small-gain theorems (see [9] for results based on trajectories and [10] for an equivalent Lyapunov interpretation).These results can be intuitively understood, from a signal perspective, as requiring that the signals be not amplified while flowing through the interconnection; and, from an energy perspective, as requiring that energy do not accumulate via the interaction between the node subsystems.
Compared to the early results focusing on single-cycle interconnection, many subsequent works have taken more general network structures into account to extend the small-gain results to large-scale interconnected systems.These works, based on the formulation of small-gain conditions, can be categorized into two approaches: 1) matrix-operator-based ones; and 2) cycle-based ones.The works in the first approach (see, e.g., [11], [12], and [13]) exploit a nonlinear counterpart of Perron-Frobenius theorem and conclude matrix operator-based small-gain conditions.These results have recently been extended to dynamical networks with infinitely many nodes in [14] and [15].The matrix-operator-condition proposed in these works can be viewed as a generalization of the spectral radius condition in the linear case, and it is equivalent to the simple contraction condition (i.e., the loop gain operator is smaller than identity) in the classical two-node case.It should however be noted that the matrix operator is composed of nonlinear functions and the conditions based on such an operator are in general difficult to check [5].
The other approach adopts graph-based small-gain conditions based on the loop gain along every cycle path in the network, which is referred to as the cyclic small-gain theorem [16].In the second approach, [17] and [18] used a max-type Lyapunov function construction and consider a max-type aggregation of the neighbours' inputs, which yields a simple contraction condition similar to the classical two-node case.The authors in [19] uses a sum-type Lyapunov function construction and considered sum-type supply rates, which also leads to a similar contraction condition but requires a decomposition of the gain functions in the node dissipation inequalities.This is because the gain functions in the sum-type dissipation inequality are associated with the edges of the underlying graph and each edge may be shared by multiple cycles.As a result, a claim on the gain of one cycle needs to use the decomposed components of gain functions to avoid overcounting.It is worth mentioning that the two approaches are closely related and the small-gain condition in one approach can be interpreted in the other approach when one uses the "max" function to characterize the influence from other nodes and to construct the network storage function (from node storage functions), see, e.g., [12] and [20].
The aforementioned results focus mostly on network analysis, whereas they have several disadvantages for control synthesis.First, most of the existing results exploit certain types of stability properties of node systems, either ISS or iISS, which excludes some systems commonly seen in practice, e.g., systems equipped with adaptive controllers.It is however possible to exploit properties weaker than node ISS/iISS to establish asymptotic stability of the interconnected system [21] or use invariance-like analysis to establish convergence of signals [22].Second, the construction of the network storage (or Lyapunov) function is typically of the "max-type".Such construction makes the statement of small-gain conditions concise, but makes the storage function nonsmooth (which requires a special treatment to derive a dissipation inequality) and has some restrictions when concluding certain stability properties [23].Third, the cyclic small-gain condition, though concise in expression, is stated cycle-wise and therefore, does not directly address the intercycle relations, e.g., common nodes and gains shared by different cycles.The latter information is especially important for control synthesis, since the control design typically affects directly the damping of a node, which further alters the gains of multiple cycles containing this node simultaneously, instead of the gain of an individual cycle.
In light of this and based on our preliminary work [24], in this article we discuss the conditions (called the small gain-like conditions) on the algebraic properties of the node dissipation inequalities and on the structure of the network that guarantee the existence of a scaling (linear or nonlinear in the sense of [25] and [26]) such that the network dissipation inequality possesses the desired algebraic properties (called the small gain-like properties), without referring to stability properties explicitly, though by adding proper restrictions to node dissipation inequalities can imply stability properties.We aim to construct a sum-type storage function for the overall network.Moreover, the supply rates for the network dissipation inequality and for the node dissipation inequalities are all of the sum type.These two features are summarized by the term "sum-type dissipation inequality" in the title.The main results contain both matrix-based and graph-based representations, depending on which one is more favourable: From an analysis perspective, we provide small gainlike conditions based on matrices (or matrix-like expressions in the nonlinear case); and from a synthesis perspective, we derive graph-based easy-to-check conditions from the matrix-based conditions with the help of the notion of active nodes.In particular, we borrow the concept of feedback vertex set [27] from graph theory to address the placement of the active nodes, which efficiently uses the "common" nodes that simultaneously belong to multiple cycles, to avoid a cycle-by-cycle gain design.We also exploit the fact that the proposed scheme does not require ISS/iISS nodes to implement an adaptive design so that detailed network parameters are not required a priori.
The rest of this article is organized as follows.Section II provides an analysis condition and a synthesis condition for a class of network systems with quadratic supply rates.Section III extends the previous results to systems with a more general class of sum-type dissipation inequalities.Section IV discusses an intermediate class of problems to achieve a tradeoff between generality and implementability.Section V discusses the control synthesis methods derived from the aforementioned results, which relate to the placement and the parameter computation/estimation of the active nodes.Section VI provides a public-health related control example to demonstrate a possible application of the proposed scheme.
Notation and preliminaries: The article uses standard notation unless stated otherwise.For an n-dimensional vector v ∈ R n , v > 0 means that v is element-wise positive and similarly for other inequality signs.An all-one (all-zero) matrix of some given dimension is denoted by 1 (0 ).For a matrix M , (M ) i denotes the ith column and (M ) ij denotes the ith element on the jth column, M > 0 means that M is element-wise positive and similarly for other inequality signs.In a directed graph, P i and S i denote the index set of the direct predecessors and successors, respectively, of the vertex i.The operator "•" indicates function composition.|S| denotes the cardinality of the set S.
The following definitions and theorem are recalled since they are useful in the remainder of the article.
Definition 1 (Z-matrix): A matrix A is called a Z-matrix (or negated Metzler matrix) if all its off-diagonal elements are nonpositive, i.e., (A) ij ≤ 0, i = j.
Definition 2 (M-matrix): A matrix A = B + sI, where B is a square Z-matrix and s is a real number not smaller than the spectral radius of B, is called an M-matrix.
Theorem 1: Let A be a Z-matrix.Then, the following conditions 1 are equivalent.
1) A is a nonsingular M-matrix.
2) A is a nonsingular M-matrix.
3) All principal minors of A are positive.4) All leading principal minors of A are positive.5) A −1 exists and A −1 ≥ 0. 6) There exists a vector v > 0 such that Av > 0.

II. SYSTEMS WITH QUADRATIC SUPPLY RATES
Consider a network of n interconnected dynamical systems in which each node, a dynamical system denoted as with respect to a storage function where n i is the dimension of the state vector of Σ i , a i > 0, and b ij ≥ 0. The nodes are interconnected via the equations u ij = y j , for all b ij = 0, whereas b ij = 0 means that Σ j is not connected to Σ i , i.e., u ij = 0.Then, the node dissipation inequalities can be written into the compact form where 1 These are extracted from 40 equivalent conditions listed in [28], in which the proof of this result is available. 2A dissipation inequality is obtained by separating x i (or y i ) from u i in the , where f i is the state evolution mapping and u i stands for all inputs to Σ i .A common technique to do this is to use the Young's inequality.See, e.g., [29] for some examples.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.and The matrix B can be understood as a weighted adjacency matrix of the underlying directed graph that describes the network structure.For example, consider a network system with the node dissipation inequalities described by the matrix The underlying directed graph is the one shown in Fig. 1.
In practice one is interested in the dissipation inequality for the overall network system.More specifically, one may wonder whether there exist positive scaling coefficients c 1 , . . ., c n such that the storage function of the overall network system, defined by where c [c 1 , . . ., c n ] satisfies the dissipation inequality V ≤ 0. In addition, in the case in which the V i 's are positive definite and radially unbounded one may require that where From the dissipation inequality (7) one can conclude Lyapunov stability of the equilibrium of the overall system and convergence of all y i 's to 0 by means of some invariance-like analysis.
To avoid results being restricted to some specific stability properties, we call the property that "there exists a construction of the network storage function as in (6) such that the network dissipation inequality (7) holds" the small gain-like property.
The aim of the article is as follows.
1) Analysis: To study the condition of the network parameters (specified by E) for which the small gain-like property holds, referred to as the small gain-like condition.

2) Control synthesis:
To use the structure of the network and to determine adjustable network parameters such that the small gain-like condition holds.To this end, the next theorem provides a small gain-like condition for the network system characterized by (2).It is worth noting that the small gain-like condition in Theorem 2 is not universal: It is straightforward to build networks for which it is not satisfied.One such network is a single-cycle interconnection containing two scalar nodes that violates the small-gain condition, i.e., either the condition   (The counterpart of this small-gain condition for more complex networks is precisely condition 4) of Theorem 1.) The small-gain analysis for the considered example reveals the fact that if one is allowed to adjust the coefficients a i 's arbitrarily one can always enforce the dissipation inequality (7), provided there is a decentralized controller on each of the nodes of the network to make the a i 's adjustable design parameters.In practice, however, this is not always feasible, e.g., because of dynamics that cannot be controlled or economic concerns that do not allow using a decentralized controller at each node.This highlights the fact that, even if Theorem 2 provides a tool for network analysis, we need to answer a question from a control synthesis perspective: How many controllers are needed to enforce the dissipation inequality (7) and where these should be placed considering the structure of the network?To answer this question we define a special class of nodes.
Definition 3: A node Σ i is called an active node if it satisfies the dissipation inequality (1) with an adjustable a i ∈ [a i , +∞), with a i > 0. 3 The property "Σ i is active" is denoted by i ∈ I A , where I A is the index set of the active nodes.
We now make a convention for graphic representation: an active node is represented by a solid green circle (e.g., nodes Σ 1 and Σ 4 in Fig. 1) and an inactive node is represented by a red dashed circle.Exploiting the concept of active nodes we now present a feasibility condition for the considered design problem.
Theorem 3: The matrix (3) can be made a nonsingular Mmatrix by adjusting the parameters a i A , i A ∈ I A , if every directed cycle of the underlying directed graph describing the network contains at least one vertex associated with an active node.
To prove Theorem 3, we need to first identify a directed cycle in the graph.It turns out that it is much easier to identify a 3 a i is a lower bound for the adjustable region of a i .This is typically fixed as it is related to the natural damping of the node system.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
directed graph without any directed cycle, i.e., a directed acyclic graph (DAG).Although most of the criteria for determining DAGs are algorithm-based, we give an algebraic condition on the relation between a DAG and the matrix (3).
Proof: First note that the matrix E associated with a DAG is similar to a triangular matrix via a permutation (which is known as topological sorting [30]), represented by a matrix P , and thus det(E) = det(P −1 )( n i=1 a i ) det(P ) = n i=1 a i .This implies that if any subgraph is a DAG the corresponding principal minor cannot contain b ij terms.Thus, b ij terms in the principal minors are all contributed by the directed cycles.Then, we use condition 3) of Theorem 1 to prove the claim.Consider the largest principal minor det(E) and note that det(E) = i∈I a i + l max l=1 ( i∈I\C l a i Π b ), where I is the set of all node/vertex indices of the graph, the structure of which is specified by E, C l is the lth set of the node/vertex indices of a directed cycle or of a union of disjoint directed cycles, l max is the total number of such sets, Π b s l j,k∈C l ,k∈S j b kj , and s l is the sign of the cofactor related to each C l .Therefore, det(E) can be rewritten as det(E) = l max l=1 ( i∈I\C l a i ( 1 It is easy to see that since there is at least one active node in each directed cycle or, equivalently, there exists i A ∈ C l such that we can always satisfy the condition 1 l max j∈C l a j + Π b > 0 by selecting one of the a i A 's sufficiently large, one can guarantee the condition det(E) > 0. Since this analysis can also be applied to the submatrix related to each of the principal minors of E, there has to be a selection of a i A for each active node such that all principal minors of E are positive.This proves that E is a nonsingular M-matrix and completes the proof.
To illustrate the result expressed by Theorem 3 consider the example illustrated in Fig. 1.The matrix E related to this graph is a nonsingular M-matrix provided a 1 and a 4 are sufficiently large, as nodes Σ 1 and Σ 4 are active node, and their associated vertices are contained in every directed cycle of the graph.Therefore, Theorems 2 and 3 reveal that as long as there is at least one active node in every directed cycle, one can adjust a i A 's such that the small gain-like condition and property hold.

III. SYSTEMS WITH NONLINEAR SUPPLY RATES
The systems discussed in Section II have dissipation inequalities with quadratic supply rates.The advantage of this formulation is that each term in the dissipation inequality is one-to-one related to a node subsystem, which allows carrying out analysis and design via the underlying directed graph.However, this puts restrictions that do not allow the use of many common nonlinear control design techniques, e.g., the use of nonlinear damping.Although the quadratic supply rate case does take nonlinear dynamic systems into account, it is natural to ask whether the results developed in Section II can be applied to a more general class of nonlinear supply rates that are not necessarily quadratic.In light of this, the rest of this section generalizes the matrix-based conditions discussed in Section II for network systems with sum-type nonlinear supply rates. 4We consider an n-node network with node dissipation inequalities given by where α i ∈ K ∞ and β ij ∈ K ∞ .This can be understood as a more general version of (8) and the gain functions β ij 's provide a more general form of the b ij in (4) which also describe an underlying weighted directed graph.This allows discussing the dissipativity properties of the network systems by discussing the weight properties of the underlying directed graph.In the case of nonlinear supply rates the constant scaling technique is not applicable in general since one cannot separate β ij and V j from the β ij • V j term which is state-dependent, whereas state-dependent scaling method [25], [26] is applicable.The first aim of this section is to find a small gain-like condition on α i and β ij that guarantees the existence of continuous functions such that the time derivative of One can see that this method is called state-dependent scaling because the construction of the overall dissipation inequality ( 11) allows λ i (V i ) to scale the supply rate to the node Σ i in a similar way as c i works in the quadratic case.We proceed by removing the coupling between λ i (V i ) and β ij (V j ) since they are depending on different variables, namely V i and V j .To do this, first recall the following result.
Lemma 1 (Young's inequality [31]): Consider a continuous strictly increasing function f and positive real numbers, a ∈ holds and in particular, equality holds if and only if b = f (a).
By Lemma 1, one can write ds, which removes the coupling between the V i -dependent λ i and the V j dependent β ij .This allows deriving a sufficient condition for (12) to hold, namely for is treated as the provided damping, then ri (V i ) ds can be treated as the demanded damping.Therefore, the implication from ( 14) to ( 12) can be interpreted as follows: If each node system dissipates the energy that it injects to its successor node systems, then the overall network system is dissipative.Consider now two types of vertices (using the index i for illustration). 1) |P i | = 0, i.e., the vertex i has no incoming edges.This means λ i can always be selected as a sufficiently "large" function so that ( 14) is satisfied, since r i (λ i , α i ) is not upper-bounded in λ i if p i = 0.Then, the vertex i can be deleted from the graph as the information of the vertex i is not needed in the analysis of the rest of the graph; 2) |P i | ≥ 1, i.e., the vertex i has at least one incoming edge.In this case r i (λ i , α i ) is upper-bounded in λ i due to the negative term −p i λ i (V i ) 0 f i (s)ds and therefore, (14) can only hold under a small-gain condition on α i and β ij .In light of these observations, we can recursively delete the vertices with p i = 0 and the outgoing edges attached to these vertices until we obtain a graph in which each vertex has p i ≥ 1; such deletions do not alter the small-gain condition we intend to derive.Thus, without loss of generality, we can consider a graph with p i ≥ 1, for i = 1, . . ., n, from the beginning.
To proceed, multiplying 1 p i by both sides of ( 15) and invoking Lemma 1 yields 1 ds and the equality holds, if and only if f with the maximizer Substituting this maximizer into (14) yields for i = 1, . . ., n, where δ i ∈ K ∞ , and the transform 5 T i is defined by We now provide two lemmas which are useful in the subsequent analysis.
Lemma 2: Consider a convex function g : R → R such that g(0) ≤ 0 and two positive real numbers Proof: By the super-additivity of the convex function g for any positive real numbers x1 and x2 , g(x 1 + x2 ) ≥ g(x 1 ) + g(x 2 ).Hence, substituting x1 = x 1 and x2 = x 2 − x 1 yields the claim.
Before proceeding, note that T i is a convex function, since is strictly increasing as required by the condition in Lemma 1.We treat convexity as a constraint on T i to be enforced later.To introduce a recursive algorithm to "solve" for T i , we rewrite ( 21) using a matrix-like representation and take a fully connected three-node system as an example, i.e., ⎡ Applying Gaussian elimination to the first element of the second and the third column using right composition (instead of multiplication) yields where due to left distributivity of multiplication, but since this does not hold for general composition operation (i.e., in general Suppose that there exists α2 such that α2 (22) with some α 2 ∈ K ∞ such that α 2 ≤ α 2 and, by Lemma 3, we can rewrite (23) as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where Exploiting the condition that α2 ∈ K ∞ , we can continue to apply Gaussian elimination to the second term on the third column.The resulting matrix is where assume that this is possible for the time being), where It is claimed that as long as there exist K ∞ -class functions α1 , α2 , α3 that bound α1 , α2 , α3 , respectively, from below, one can con- A simple way to see this is to solve (26) for T 1 , T 2 , T 3 in terms of δ 1 , δ 2 , δ 3 .These allow writing T 1 , T 2 , T 3 as positive sums of composition terms led by δ 1 , δ 2 , δ 3 .By properly selecting δ 1 , δ 2 , δ 3 , these positive terms can be individually made convex, and the convexity is preserved under positive summation, which further justifies the use of Lemma 1. Finally, using ( 17) and ( 19) yields which is the state-dependent scaling function needed.For large-scale systems it is not practically useful to derive a closed-form expression for αi , as it can be shown that the total number of the composition terms in αi is described by the Sylvester's sequence [34] subtracted by one 6 , which grows doubly exponentially with n, the number of node systems.Instead, we seek a recursive algorithm that equivalently performs iterative Gaussian elimination (as demonstrated above) and at the same time reveals the underlying topological meaning of the terms in αi .This recursive algorithm is described in Algorithm 1, the main idea of which is that: Each entry, indexed by (i, j), of E T (the lower triangular matrix defined by ( 26)), contains the composition terms formed by a term from an upper entry (k, j) in the same column, the inverse of αk (which is the lower bound of αk ), and a term from a left entry (i, k) in the same row, 6 To see this, first note that the expression of αi is given by a sumof-composition formula.Let N i denote the number of composition terms.Then, one can find that N 1 = 1 and The upper triangular entries are eliminated 5: else if i = j then 6: αi The diagonal entries are expressed as T i • 1 p i αi due to Lemmas 2 and 3 10: else 11: Compute the lower triangular entries without merging the T i terms sequentially composed from right to left, and such a construction holds recursively.From the aforementioned iterative Gaussian elimination procedures, it is not difficult to see the reasoning behind: Each term in the (i, j) entry was brought from the (i, k) entry as a side effect of eliminating the (k, j) entry using αk .The computation of (σ n ) k is straightforward once αk has been computed.Since (σ n ) k is derived by the same Gaussian elimination procedure as the one that derives αk , (σ n ) k is simply the positive sum of each composition term in αk , after removing the sign and replacing the first function of each composition term with the δ l , where l is the column index of the first function.
With the help of the recursive algorithm, we can conclude the following small gain-like result.
Theorem 4: Consider a network system with node dissipation inequalities given by (9).If there exists αi ∈ K ∞ such that αi ≤ αi , for i = 1, . . ., n, with αi computed by Algorithm 1, then there exist continuous functions λ i : R ≥0 → R ≥0 , i = 1, . . ., n such that (10) holds, and the overall dissipation inequality defined by (11) satisfies for Proof: Given the existence of the K ∞ -class functions αi , i = 1, . . ., n, Algorithm 1 is feasible and its output is a lowertriangular matrix E n T and the functions αk , which further yield the functions (σ n ) k .For ease of expression, we drop the superscript n without causing ambiguity since all the variables used have the same superscript.Let σ denote From the previous analysis we know that Each element in σ is a positive sum of composition terms, with each composition term led by one of the δ i , i = 1, . . ., n.In addition, the off-diagonal elements of E T are negative sums of composition terms led by one of the Ξ ← (E T ) ij ξ stores the expression to be returned 6: for k = 1 to min(i − 1, j − 1) do 7: Ξ U : expression of the (k, j)-entry (upper expression), with T k and the minus sign of each term removed 10: Find αk ∈ K ∞ such that αk ≤ αk 13: for each ξ U in Ξ U do 14: for each ξ L in Ξ L do 15: This accounts for the effect of eliminating the (k, j)-entry on the (i, j)-entry 16: return Ξ 17: function REMOVET (i, Ξ) 18: for each ξ in Ξ do 19: remove T i from ξ 20: return Ξ T i 's, thus T 1 , . . ., T n resulting from (30) can be written as positive sums of composition terms led by one of the δ i 's.Note that to meet the conditions of Lemma 1, T 1 , . . ., T n are required to be strictly convex: This is to guarantee that their derivatives f −1 1 , . . ., f −1 n are strictly increasing.This is enforced by properly selecting δ i ∈ K ∞ such that each individual composition term is strictly convex.Since convexity is preserved under positive summation, the resulting T 1 , . . ., T n are also strictly convex.Compute λ i (V i ) using (28), and consider the overall storage function (11).By Lemma 3, Remark 2: The small gain-like condition based on αi is purely determined by the coefficients in the node dissipation inequalities and the network topology.In contrast, existing results considering both sum-type construction of storage function and sum-type supply rates either require the existence of λ i as a part of the small-gain condition [13] or require decomposing the original network gains prior to formulating the small-gain condition [19].
Remark 3 (Conservativeness): To discuss the conservativeness of the condition in Theorem 4, we first consider the relationship between such a condition and the classical small-gain condition for two-node systems.One can regard αi ≥ αi ∈ K ∞ as the counterpart of the small-gain condition for network systems since such a condition guarantees the small-gain property, i.e., the existence of λ i such that V < 0. Take the classical two-node feedback interconnection as an example, where, ac- We can show that the condition in Theorem 4 is equivalent to the sufficient and necessary7 small-gain condition [36, (34)], namely where ω i ∈ K ∞ , i = 1, 2, and Id is the identity mapping.To see this, first note that α1 ≥ α1 always holds by selecting α1 α 1 .
We then show that existence of α2 ∈ K ∞ such that α2 ≥ α2 implies (31) Therefore, the left-hand side of ( 31) is less or equal to (31).
To show that (31) implies the existence of α2 , note that (31) , which yields a valid α2 and proves its existence.Hence, the equivalence is proven.In this sense, the condition in Theorem 4, for two-node systems, is no more conservative than its counterpart in [36], though the small gain-like properties pursued by this article are different from the robust version discussed therein, and thus the necessity does not hold in this article even for the two-node case.
It is worth mentioning that the two-node case can be extended to single-cycle interconnection containing arbitrarily many nodes because one can recursively combine cascade nodes into one node until the network is reduced to the feedback twonode case.For denser networks with nodes having more than one input, the proposed Gaussian elimination-like analysis results in conservativeness.This can be seen from how we decouple the β ij terms from the λ i terms when deriving (14): Despite β ij 's being distinct functions for different j, the "scaling" function of Young's inequality is the same f i .This means that unless the network satisfies certain symmetry, the equality sign in Young's inequality does not hold collectively.The common "scaling" function also causes the separation of p i and 1 p i in (18), which does not allow the cancellation of each other in the presence of T i and yields a restrictive condition when the network is large and dense due to large p i .It turns out that using distinct f ij for each associated β ij does not allow using a single T i in each row in the Gaussian elimination-like procedure.The relaxation of such a restriction will be studied in future work.
Similarly to Definition 3, one can define the notion of active nodes, provided a node has an adjustable α function.
Definition 4: A node Σ i is called an active node, denoted i ∈ I A , if it satisfies the dissipation inequality (9) with an adjustable α i ∈ K ∞ , and α i ≥ α i , with α i a positive definite function 8 .
It turns out that one can select the α i A , i A ∈ I A , to let the algebraic condition of Theorem 4 hold, and the feasibility of this is closely related to the location of the active nodes in the network system, even though the condition and the proof of Theorem 4 are completely different from Theorem 2. In other words, there is a nonlinear counterpart of Theorem 3, as stated below.
Theorem 5: The condition of Theorem 4, namely, the existence of αi ∈ K ∞ , holds for some admissible class K ∞ functions α i A , i A ∈ I A , if every directed cycle of the underlying directed graph contains at least one vertex associated with an active node.
Proof: To prove the claim note the graph-theoretic interpretation of the COMPUTEENTRY function of Algorithm 2: For each vertex i, the expression of αi is obtained from α i subtracting several composition terms, with each composition term associated to one of the closed paths starting from vertex i and returning back to vertex i, regarding the off-diagonal elements of E I T as an analogy to the adjacency matrix.The upper expression ξ U contains the outgoing path, and the left expression ξ L contains the incoming path, and such interconnection is made recursively.In this sense, αi can written as where and k l j ∈ C i l , C i l is the index set of the lth circuit path 9 from vertex i to itself, in the subgraph consisting of vertices 1 to i and the associated edges; l max is the total number of such sets; m |C i l |; and k l j ∈ S k l j−1 .We proceed by noting that if α i is adjustable in the sense stated in Definition 4, then αi is also adjustable in the same sense and so is αi .This is true because one can always use α i to dominate γ l 's in (32).Since γ l 's are associated with circuit paths, having at least an active node in every cycle implies that there is at least an active node in every circuit, and therefore, γ l 's can be made arbitrarily small by adjusting the active α−1 i A function.Thus, any α i ∈ K ∞ can dominate αi and guarantee the existence of αi ∈ K ∞ such that αi ≤ αi , for i = 1, . . ., n.Hence, the conditions of Theorem 4 hold, which completes the proof.

IV. LINEARLY PARAMETRIZED SUPPLY RATES
Section III has shown that the notion of active nodes is applicable to nonlinear systems with sum-type dissipation inequalities and supply rates that are not necessarily quadratic.While on one hand this extends the results of Section II to more general scenarios, on the other hand it makes the formulation of the damping functions α i abstract, and therefore difficult to be used in practical designs.Thus, in this section we consider a "middle ground" between the schemes discussed in Sections II and III which allows parametrizing the supply rates and allows the implementation of standard nonlinear control design techniques that are not well catered by the quadratic supply rate case.
To start with, consider a two-node system with storage functions 9 The term "circuit path" is used here since the path may not be a simple circuit, or equivalently a cycle: it can contain nested cycles.See (27) for an example: the longest term indicates the path 3 → 1 → 2 → 1 → 3, which contains two cycles.
where y 1 ∈ R, y 2 ∈ R. Obviously, the selection φ(y) = [y 2 1 , y 2  2 ] in the spirit of Section II does not work for this system since it does not take the y 4  1 -terms into account.In the spirit of Section III one can write α 1 (s) = a 1 s + a 3 s 2 , α 2 (s) = a 2 s, β 12 (s) = b 12 s, β 21 = b 21 s, which properly describes the system.Nevertheless, the parametrization of α 1 , even though a 1 and a 2 can be selected arbitrarily, does not fulfill the condition for an active node specified by Definition 4. This suggests finding an alternative formulation to allow exploiting parametrization of node dissipation inequalities.Alternatively, based on the formulation in Section II, we could augment φ(y) with an extra positive definite term, in this case, y 4  1 , and define the augmented vector φ(y) = [y 2  1 , y 2 2 , y 4  1 ] .Compared to the case in Section II, in which each term of the supply rate is one-to-one related to a node, now y 2 1 and y 4 1 are both related to the same node, node Σ 1 .In other words, node Σ 1 has two supply rate basis functions (basis functions for short), y 2  1 and y 4 1 , while node Σ 2 has only one basis function y 2  2 .To generalize this idea suppose that there are q i basis functions for the dissipation inequality of node Σ i and q n i=1 q i basis functions in total.Define φ(y) = [φ (y), ϕ 2  1 (y 1 ), . . ., ϕ q 1 1 (y 1 ), . . ., ϕ 2 n (y n ), . . ., ϕ q n n (y n )] ∈ R q , where ϕ j i (y i ) denotes the jth basis function of node Σ i , and φ(y) = [ϕ 1  1 (y 1 ), . . ., ϕ 1 n (y n )] ∈ R n is the vector containing the primary basis functions10 of each node, with the other elements in φ(y) referred to as secondary basis functions.Under this definition of φ(y), the node dissipation inequalities are with a k i > 0, b k ij > 0, and these can be written into a vector form similarly to (2), namely where Ē ∈ R n×q contains the coefficients of the node dissipation inequalities with opposite signs.With the same construction of the overall storage function as (6), the overall dissipation inequality can be derived as V ≤ −c Ē φ.Define the left n × n submatrix of Ē as E, and the right n × (q − n) submatrix as Ẽ.
For example, the matrix Ē associated with the two-node system with dissipation inequalities ( 33) is To make the notation natural in the matrix Ē we remove the superscript of the coefficients in the matrix expression (34) and replace the subscript of a (•) and the second subscript of b (•) (indicating the predecessor node) with the index of the corresponding basis function in φ(y) (or equivalently, the column index of the coefficient in Ē).This defines an alternative set of indices î(i, k) for coefficients associated with the kth basis function of node Σ i .For example, in (36) we have î(1, 2) = 3 for the second basis function of node Σ 1 and therefore we replace a 2 1 with a 3 and replace b 2  21 with b 23 .Since Ē is not a Z-matrix, if we want to use the results established in Section II it is better to restore the one-to-one relation between each basis function and each matrix dimension by augmenting Ē so that we have a Z-matrix to analyze.From a graph theoretic perspective, this requires adding some augmented vertices to the underlying directed graph so that the total number of vertices is the same as the basis functions rather than the number of nodes.To this end, rewrite the overall dissipation inequality as where Ê is an q × q Z-matrix augmented from Ē, and ĉ ∈ R q is augmented from c.More specifically, Ê can be written as where U ∈ R n×(q−n) is defined as In other word, the underlying graph described by Ê is obtained by adding vertices (indexed by î) associated with the secondary basis functions, and then by connecting these augmented vertices to the graph described by E according to the dissipation inequalities, which is associated with the primary basis functions and the original network.Since the scaling operation is implemented on the n node systems, not on the q − n augmented vertices, as they do not originate from new node systems, the last q − n elements of the augmented scaling vector ĉ are generated by the first n elements, i.e., the original scaling vector c.It is not difficult to show that this fact can be described by the constraint where L ∈ R (q−n)×n is defined as Consider again the two-node system (33).Then, U = [1, 0] and the augmented version of ( 36) is Since the third column of M comes from the first dissipation inequality, which should be multiplied by c 1 , we define ĉ To derive the counterpart of Theorem 2 we have to first answer two questions: 1) how to determine whether Ê is a nonsingular M-matrix by checking the original matrix E; and 2) how to use condition 6) of Theorem 1 considering the additional constraint (40).The answer to the first question is given by the result below.
Lemma 4: Ê is a nonsingular M-matrix, if and only if E is a nonsingular M-matrix.
Proof: Consider the block triangular structure of the matrix Ê defined by (38).The leading principal minors with order higher than n have the same sign as det(E).Therefore, using condition 4) of Theorem 1 completes the proof.
To answer the second question we need the problem we are trying to deal with to be clarified.Condition 6) of Theorem 1 Fig. 2. Underlying directed graph specified by (42).The notation "3 ← 1" means that vertex 3 is an augmented vertex which originates from node Σ 1 .
guarantees the existence of ĉ > 0 such that Ê ĉ > 0 if Ê is a nonsingular M-matrix, which, however, is not sufficient in this case as there is the additional constraint (40) on ĉ.Thus, we need to add an additional condition to Ê such that ĉ also satisfies the constraint (40).This leads to the following result.
Having proved Lemmas 4 and 5 we can proceed to give a criterion for the existence of scaling coefficients such that the dissipation inequality ( 7) holds.
Theorem 6: Consider the node dissipation inequalities given by (35).There exists a vector of scaling coefficients c > 0 such that V constructed as in ( 6) satisfies the dissipation inequality (7) if both the following conditions are satisfied: 1) the n × n leading principal submatrix of Ê is a nonsingular M-matrix; and 2) there exists σ > 0 such that σ is in the kernel of L( Ê ) −1 .
Theorem 6 reveals the fact that the n × n leading principal submatrix of Ê plays an important role.It is easier to understand this from a graph-theoretic perspective.Since all the augmented columns in Ê have only one nonzero element on the diagonal entries, the associated vertices in the underlying directed graph have only outgoing edges, but no incoming edges (see Fig. 2), which guarantees that no directed cycle contains these augmented vertices.In other words, the augmentation of the graph does not create new directed cycles and all directed cycles in the graph are specified by the n × n leading principal submatrix of Ê.
Note now that each node can have more than one basis function, and therefore we need to slightly extend Definition 3.
Definition 5: A node Σ i is called an active node, denoted by i ∈ I A , if it satisfies the dissipation inequality (34) and for k = 1, . . ., q i , the damping coefficients a k i are adjustable in [a k i , +∞), with a k i > 0. The indices of all vertices (including the augmented vertices) of the underlying graph that originate from active nodes make up the set ÎA .
This definition allows the damping coefficients a (•) of all vertices (including the augmented vertices) in the underlying directed graph that originate from active nodes, to be adjustable.For instance, in the underlying directed graph of the two-node example, both vertex 1 and vertex 3 originate from node Σ 1 , and thus both a 1 and a 3 are adjustable if node Σ 1 is an active node.Having clarified this, we are ready to see how to enforce the dissipation inequality (7) with active nodes.
Theorem 7: For all σ > 0 there exists a selection of a k i A , i A ∈ I A , k = 1, . . ., q i A , and a vector of scaling coefficients c > 0, depending on σ, such that V constructed as in ( 6) satisfies the dissipation inequality V ≤ −σ φ(y) if both the following conditions are satisfied: 1) every directed cycle of the underlying directed graph describing the network contains at least one vertex that originates from an active node; 2) every augmented vertex originates from an active node.
Proof: We first consider the original n-vertex graph without the augmented vertices.Condition 1) of this proposition and Theorem 3 indicate that for all σ > 0, there exists a selection of a 1 i A , i A ∈ I A , i.e., the damping coefficients associated with the primary basis function of the active nodes, and c > 0, depending on the choice of σ, such that the n × n leading principal submatrix E satisfies E c = σ.Define ĉ [c , c ] , with c ∈ R q−n to be determined.Note that ( 40) and ( 41) yield U c − c = 0, or equivalently, c = U c. Since U > 0 and c > 0, we have c > 0, which provides a valid candidate for ĉ > 0.
We proceed to prove that for such a ĉ we can select a k i A , k = 2, . . ., q i A , i A ∈ I A , i.e., the damping coefficient of the secondary basis functions of the active nodes, such that the claim holds.To see this, note that where Ê12 U diag(a n+1 , . . ., a q ) − Ẽ is independent of the a (•) coefficients (cancelled by the subtraction) and only depends on the b (•) coefficients, and therefore Ê12 ≥ 0. Thus, we have − Ê 12 c + [a n+1 , . . ., a q ]c = σ, or equivalently, [a n+1 , . . ., a q ] = (diag(c)) −1 (σ + Ê 12 c), for all σ > 0, which gives a vector of positive candidates for a n+1 , . . ., a q .It remains to check whether these a (•) are in the interval specified by Definition 5.If the resulting a (•) is below the lower bound, we replace the value with the lower bound.This guarantees that Ê ĉ ≥ σ, and therefore V ≤ −ĉ Ê φ(y) ≤ −σ φ(y).Combining the arbitrariness of σ > 0 and σ > 0, we conclude that σ = [σ , σ ] > 0 is also arbitrary.Hence, the proof is complete.
The augmented vertices provide a convenient tool for network stabilization via nonlinear damping, in which case the augmented vertices are associated with the additional nonlinear damping terms.An illustrative example can be found in [24,Sec.IV].

V. CONTROL SYNTHESIS VIA ACTIVE NODES
We have focused on the small gain-like condition for analysis and the feasibility conditions on the location of the active nodes to enforce such small gain-like conditions.In this section, we move forward to answer the "how to" part of the problem from a synthesis perspective, i.e., the placement of the active nodes and the adjustment of their damping coefficients.

A. Placement of Active Nodes
From Theorems 3, 5, and 7 we know that the active nodes should be placed in the network such that every directed cycle in the underlying graph contains at least one of the vertices associated with the active nodes.Hence placing the active nodes, from a graph-theoretic perspective, boils down to finding a set of vertices such that after the removal of these vertices (and the edges attached to them), the remaining graph is acyclic.This set is commonly known as the feedback vertex (node) set (FVS) in the graph theory and the computing theory literature.In general, one may want to use a minimum number of active nodes to achieve the desired network dissipation inequality, which leads to the minimum FVS problem.This problem is NP-complete for directed graphs [27].Many contributions have studied the exact solution of the minimum FVS problem, see, e.g., [37], [38], and [39].We present here a method exploiting the permanent of a matrix to search for the minimum FVS.The permanent of a matrix A ∈ R n is defined as per where π is one of the permutations of the set {1, . . ., n}.A more intuitive interpretation is that per(A) can be computed by using Laplace expansion without switching the sign of the product terms as when computing det(A).As discussed in Section II, each product term in det(E) is associated with directed cycles or unions of disjoint directed cycles of the underlying directed graph of E. Note that by defining the n × n matrix Ω E as The number of Laplace expansion terms of det(E) equals per(Ω E ).It is then natural to understand that vertex i is contained in (per(Ω E ) − per(Ω Ei )) directed cycles and unions of disjoint directed cycles, where Ω Ei is the matrix obtained by deleting the ith row and the ith column of Ω E .This leads to the following lemma.Lemma 6: Consider E and its underlying graph.Let vertex i and vertex j be in the same directed cycle.Vertex i is contained in at least as many directed cycles as vertex j is, if and only if Proof: Let m i and m j be the total number of directed cycles and unions of disjoint directed cycles containing the vertex i and the vertex j, respectively.We have m i = (per(Ω E ) − per(Ω Ei )), and m j = (per(Ω E ) − per(Ω Ej )).Therefore, m i ≥ m j if and only if (45) holds.Note that m i and m j also count the number of unions of disjoint directed cycles, but since the vertex i and the vertex j are in the same directed cycle, the contribution of the considered cycle to the values m i and m j (via the considered cycle and the unions of the considered cycle with other disjoint directed cycles) are the same.Hence, m i ≥ m j if and only if the vertex i is contained in at least as many directed cycles as the vertex j is, which completes the proof.Algorithm 3: Algorithm to place the minimum number of active nodes in a given network system described by (35).Input: The dissipation coefficient matrix Ē Output: Index set of the placed active nodes I A 1: Build a directed graph G and index its vertices according to E (as per ( 36)) 2: Compute U from Ē (as per ( 39)) The for-loop below allocates active nodes to satisfy condition 2) of Theorem 7 3: for each row i of U do 4: if row i of U is not all-zero then 5: Attach i to I A 6: Delete vertex i and its attached edges from G The while-loop below solves for the minimum FVS of the remaining graph 7: while G is not acyclic do 8: for each vertex i in G do 9:

its attached edges from G
This lemma indicates that one can find the most "important" vertex in a given directed cycle by determining the vertex which has the largest decrease in per(Ω E ) after the deletion of itself and its attached edges, as this vertex is contained in more directed cycles than any other vertex in the considered direct cycle, and therefore should be included in the solution of the minimum FVS.Algorithm 3 is developed under this spirit to place the active nodes efficiently.The for-loop part of the algorithm is straightforward as this fulfills the requirement of Theorem 7. The while-loop part is essentially an exact algorithm to solve the underlying minimum FVS problem.
Proposition 1: The while-loop in Algorithm 3 finds one 11 of the minimum FVSs of G.
Proof: First note two facts: 1) G contains a finite number of directed cycles; and 2) at least one vertex in each directed cycles has to be included in the minimum FVS.If G is acyclic, the minimum FVS is the empty set, which is consistent with the algorithm as the while-loop breaks without attaching any indices.Otherwise m i A ≥ 1, and vertex i A is contained in at least one directed cycle.Due to fact 2), one has to include at least one vertex to the minimum FVS for each such cycle, and due to Lemma 6, vertex i A is the vertex such that the deletion of itself and its attached edges removes the most directed cycles in G.At each step of the while-loop, the selected vertex i A is among the vertices, one of which has to be selected, and is the one that removes the most directed cycles from the finite number mentioned in fact 1).Hence the algorithm removes all directed cycles in G with minimum steps, which completes the proof.
Algorithm 3 is at most of complexity O(p(n)2 n ) or equivalently O * (2 n ), where p(•) is a polynomial.To see this note that the complexity is dominated by the while-loop solving for the minimum FVS and, in each single loop, the complexity of the algorithm is dominated by the computation of the permanent.Computing the permanent of an n × n matrix is of O(n2 n ) using Ryser's formula in the Gray code order [40], [41], and in the worst case, the number of permanents that need to be 11 The minimum FVS of a graph is, in general, not unique.computed is a polynomial of n, which means that the overall complexity is O * (2 n ).The efficiency can be further improved if one computes the permanent using the method proposed by [42], which reduces the overall complexity to O * ((2 − ) n ), where > 0 is a constant depending on the sparsity of the graph.This is comparable to the complexity of existing exact methods, e.g., [39], for the solution of the minimum FVS problem for directed graphs, while the implementation of the algorithm is much more compatible with the design methods proposed in this article, as it is purely based on matrix computation and only uses the information from Ē.
In general, the exact methods solve for the minimum FVS in directed graphs in exponential time, which is not acceptable when the considered network is large.In practice, the number of available active nodes can also be limited (this is also why one needs to find a placement method using the minimum number of active nodes), which restricts the associated FVS problem to a parametrized FVS problem, in which the parameter is the upper bound of the cardinality of the FVS.The algorithms proposed in [43] and [44] can solve the parametrized FVS problem in linear time or report that no FVS smaller than the given bound exists.Furthermore, if we allow trading off the computation time against the number of active nodes used, the approximate method proposed in [45] can be used for computing the permanent and this reduces the computation time to polynomial in a probabilistic sense.It should be noted that such a reduction in computation time is achieved at the cost of precision and the resulting set may not be a minimum FVS as Proposition 1 may not hold, but the result is guaranteed to be an FVS by definition since the while-loop in Algorithm 3 checks whether the graph is acyclic before its termination, and therefore, it still computes a valid placement of active nodes.

B. Computation of Damping and Scaling Coefficients
Once the location of the active nodes has been determined, and the associated vertices comprise an FVS, one can solve for the damping coefficients of the active nodes and the associated scaling coefficients.In what follows, we show that the solution for these coefficients can be formulated as the solution of a set of linear inequalities, despite the nonlinear coupling between the coefficients.We consider the linearly parametrized nonlinear supply rates discussed in Section IV.Recall that for the network system with node dissipation inequalities (35) and the network storage function (6), the network dissipation inequality is (37).As a result, the dissipation inequality (7) holds if Ê ĉ > 0, for some ĉ > 0 subject to (41).Note that Ê can be decomposed as Ê = M + M , where (46) and with ãi a i − a i .Note that in M ĉ, for i A ∈ ÎA , the decision variables ãi A are multiplied by ĉi A , which are also decision variables, creating nonlinear couplings.the relation This allows defining a constant matrix such that M ĉ = N d and therefore, Ê ĉ = M ĉ + N d.This yields a system of linear inequalities and equations given by This describes an admissible region that is nonempty due to Theorem 7, yet not a direct solution for ĉ and d.In light of this, one can rewrite (50) into an optimization problem with some objective function f to be specified, namely where ]; σ > 0 is the same as the one used in Theorem 7 that specifies a minimum guaranteed dissipation rate −σ φ(y); v [ε , 0 1×| ÎA | ] , with ε > 0 to keep ĉ away from 0; v > 0 is an "upper bound" to make the admissible region compact and suitable for the use of off-the-shelf optimization solvers.Typically, one may want to keep the damping coefficients of the active nodes as small as possible while fulfilling all the design specifications.
In this case, one may consider the objective function j=1 ãi Aj or a weighted version of this.Since this cost is nonlinear and may complicate the computation, one can also consider the linear cost f (v) = w v, where w is a vector of constant weights.One only needs to tune w in such a way that w i Aj w q+j , which leads to a large ĉi Aj and a small d j , yielding a small a i Aj .This reduces (51) to a linear programming problem and allows searching for v efficiently, which is especially favourable for large-scale network systems.It should be noted that the selection of the objective function only affects the solution for ĉ and a i A , i A ∈ ÎA , whereas all solutions yield the same network dissipation inequality V ≤ −σ φ(y).
We complete our discussion by showing that Theorem 7 also guarantees that the admissible region of the reformulated optimization problem (51) is nonempty.
Proposition 2: Consider the problem (51) and assume that the conditions of Theorem 7 are satisfied.Then, for all ε > 0, there exists a v, depending on ε, such that the admissible region of (51) is nonempty.
Proof: By Theorem 7, setting a constant σ > 0, there exist ĉ > 0 and a i A > a i A , i A ∈ ÎA such that Ê ĉ = σ and Lĉ = 0. Consider now a scalar λ and a new scaling vector ĉ λĉ.Keeping Finally, note that v also satisfies the first two conditions of (51) since λ ≥ 1, and hence the admissible region contains at least v .This completes the proof.

C. Adaptation of Damping Coefficients
The result presented in Section V-B provides a solution for both the scaling coefficients for all node systems and the damping coefficients for the active nodes.From a control synthesis perspective, the computation of the scaling coefficients is unnecessary as they do not appear in the implementation, and the existence of such scaling coefficients are sufficient to establish desired stability properties.Lifting the requirement of computing the scaling coefficients allows obtaining damping coefficients for the active nodes via adaptation instead of explicit computation.In what follows, we present how to implement decentralized identifiers to achieve this.
For an active node Σ i A , i A ∈ I A , one can substitute the parameter estimates âk i A , k = 1, . . ., q i A for the damping coefficients a k i A by modifying the local control law.Define , and consider the parameter update law where Γ i A is the positive and diagonal adaptation gain matrix, and proj(θ, v) is the projection operator proposed in [46], which projects the vector v onto the tangent space (at θ) of a convex region Θ (with smooth boundary) to which θ belongs, if θ is outside Θ and v is pointing out of Θ.Therefore, the condition holds.By Theorem 7, the existence of θ is guaranteed and one can always select Θ such that it is contained in the cone defined by Definition 5 and sufficiently large to contain the true parameter θ.It turns out that one only needs to augment the node storage functions with a positive definite function of the parameter estimation error θ to obtain a result similar to Theorem 7 without the exact knowledge of θ nor the computation of it.Proposition 3: Consider the augmented node storage functions . ., q i A , replaced by the parameter estimates θi A (updated by ( 52)), and the original node storage functions with dissipation inequalities (34) for the other nodes.If the two conditions on the active nodes in Theorem 7 hold then, for all σ > 0, there exists c > 0 and convex regions of projection Θ i A , depending on σ, such that the network dissipation inequality V θ ≤ −σ φ(y) holds, where the augmented network storage function is given by V Similar to what is done in passivity-based adaptive control, 12 the parameter estimation error terms are cancelled by the parameter update law (52).This means that the node dissipation inequalities associated with the augmented storage function in the adaptive case have the same supply rates as that of the original storage functions (34) in the case in which the a k i A are known.This allows using the unknown coefficients a k i A as if they were known (the certainty-equivalence principle) in the subsequent analysis as long as their nominal value is contained in the convex regions Θ i A .This reduces the problem to the same problem solved by Theorem 7, and the rest of the proof is straightforward by noting that the existence of Θ i A is guaranteed by the existence of the nominal parameter a k i A .This completes the proof.
Remark 4: After incorporating the dynamic parameter estimate θi A into the associated active node Σ i A , neither the active nodes nor the overall network system is ISS or iISS, which makes it difficult to directly apply existing methods that require certain stability properties (e.g., [12] and [19]).In contrast, the result in Proposition 3 is still useful as we only need a supply rate with negative definiteness in y to conclude the convergence of y to zero.
Remark 5: The regions of projection Θ i A have to be determined when implementing the parameter update law ( 52): This requires some knowledge of a k i A .This is not as restrictive as the requirement to know the exact value of a k i A in two senses.First, one can make Θ i A sufficiently large to relax the requirement of such knowledge.Second, it is possible to implement the update law without projection, e.g., with adaptation gain γ k i A > 0 and initial condition âk i A (0) ≥ a k i A .In this case the parameter estimates are decoupled and each of them is nondecreasing over time.Due to the initial conditions we set, these parameter estimates satisfy the constraints of Definition 5.This is a special case of (52) in which Γ is diagonal and one can prove the counterpart of Proposition 3 in a similar approach.A numerical example using the parameter update law in (54) can be found in [24].
In addition to removing the need for explicit computation, the adaptive update laws used, either (52) or (54), only require information on the node Σ i A , which allows the implementation to be decentralized.This is more favorable in some scenarios in which the information of other nodes is unavailable, compared to the method in Section V-B.

VI. DESIGN EXAMPLE
In this section, we illustrate the proposed results using a disease control example.Consider the deterministic Susceptible-Infectious-Quarantined-Susceptible (SIQS) model [47], [48], a modification from the Susceptible-Infectious-Susceptible (SIS) model (see e.g., [49,Sec. 10.3]).The isolated version of the SIQS model is described by the equations where S(t) ≥ 0 is the susceptible population, I(t) ≥ 0 is the infectious population, β > 0 is the infection constant, γ > 0 is the recovery constant, and δ > 0 is the quarantine constant.Since Ṡ + İ + Q = 0, the total population N S + I + Q remains constant.This model describes infectious diseases that are in general nonlethal and for which immunity is not acquired after recovery, like influenza.Therefore the infectious population and quarantined population return to the susceptible population once recovered, instead of being removed.Though compartmental models (like SIQS) for isolated settlement provide a fundamental tool to understand the dynamics of a pandemic and associated small-gain-based analysis methods have been studied in, e.g., [50], we also need control synthesis methods to stop an emerging pandemic in a network of interconnected settlements, since the interaction among the settlements is one of the major driving forces of a pandemic.To this end, extend the isolated SIQS model to a scenario in which n interconnected settlements and the population migrating among them are considered.For simplicity, we assume that the total population remains constant for every settlement, i.e., for i = 1 . . .n, as the population that the facilities of each settlement can support is approximately constant in the considered time horizon.Consider now the interconnected version of the infectious population dynamics, i.e., where μ ij ≥ 0 denotes the rate of migration from settlement j to settlement i.Clearly, (56) holds, if and only if which indicates the balance of migration.Substituting (56) into (57 It is not difficult to verify that R + is forward invariant for S i , I i , Q i , and therefore P i (t) ≥ 0 for all t ≥ 0. Thus, one can select V i (I i ) = I i as the storage function for each node and the associated dissipation inequality is where It can be observed that in the case in which neither migration nor quarantine is considered (i.e., all μ-terms and δ-terms are zero), the infectious population of each settlement converges to zero if the basic reproduction number denoted by R 0i or equivalently, β i > γ i .If quarantine measures are adopted, there exists a δ i > 0 such that δ i > β i − γ i , which guarantees that the local infectious population asymptotically converges to zero.For the convenience of the subsequent discussion, we assume that the "baseline" quarantine force at each settlement is at least locally "sufficient" in the sense that holds for i = 1, . . ., n.This guarantees that a i > 0 and the convergence of the infectious population to zero if each settlement is isolated from the others (i.e., b ij = 0).However in the interconnected case, this is more complicated due to the nonzero Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.b ij -terms.The infection in the network system can be amplified through the interconnections even if the damping effect at each node is sufficient for the isolated scenario.This is one of the reasons why cutting off transport is considered a public health control measure.In what follows we show, using the notion of active nodes, that it is possible to bring the infectious population to zero at each settlement without the need for cutting off intersettlement transport if the quarantine forces at the settlements that serve as the "transport hubs" are adjustable.Now, consider the ten interconnected settlements described by the graph in Fig. 3 with the unit: thousand people/day.One can verify that the given M matrix satisfies conditions (58), for i = 1, . . ., 10.
Note that the node dissipation inequalities (59), under conditions (60), are in a similar form as (2).Instead of implementing an indiscriminately strengthened quarantine policy for all settlements, we can exploit the notion of active nodes so that only the δ i 's of the "critical" settlements specified by an FVS are required to be adjusted.For a network of complexity as depicted in Fig. 3, it is difficult to find a minimum FVS manually.To this end, we implement Algorithm 3 and obtain that I A = {1, 2, 9}.The next step is to compute the quarantine forces δ 1 , δ 2 , and δ 9 .Consider now three scenarios for a three-month period with different quarantine policies.Scenario 1, the control group, adopts the "baseline" indiscriminate quarantine policy with identical quarantine force δ i = 0.5 at each settlement.Though condition (60) holds for all nodes, infectious populations do not converge to zero, as shown in Fig. 4, Scenario 1.The infectious population and quarantine population reach a nonzero equilibrium, which is also called the endemic steady state (see, e.g., [52,Ch. 7]), a common phenomenon of SIS or SIQS models.The other two scenarios are the experimental groups.In Scenario 2, we compute the quarantine forces in the spirit of Section V-B by solving the optimization problem (51), with the objective function f (v) = d d and the box constraints v = 101 , v = 10 2 1 .This yields δ 1 = 1.39, δ 2 = 1.36, and δ 9 = 1.34.In Scenario 3, we exploit the adaptation mechanism in Section V-C, i.e., dynamically estimate the required quarantine forces, using the equations δi A = I i A , δi A (0) = 0.5, i A ∈ I A .Note that due to the monotonicity of the estimated quarantine forces and the given initial conditions, the projection operation is not needed in this scenario.Invoking Proposition 3, the supply rate to the network dissipation inequality is negative definite in I and applying standard invariance analysis yields the convergence of I to zero.The simulation results shown in Fig. 4, and Scenarios 2 and 3, verify the convergence of infectious and quarantined populations to zero in both scenarios.
Remark 6: Comparing Scenarios 2 and 3 in Fig. 4, one observes that the computation-based quarantine policy is more effective than the adaptation-based policy in the early stage.This is because in Scenario 2 the δ i 's are computed offline, and thus the quarantine forces are "sufficient" from the beginning.The adaptation-based policy, though acts slower in the early stage, is decentralized in the sense that it requires only local infectious populations rather than detailed migration data of the entire settlement network.Therefore, it is more useful when accurate global data is not available to all policy-makers.

VII. CONCLUSION
The article has introduced the notion of active nodes, i.e., node subsystems with adjustable damping coefficients in the node dissipation inequalities.Based on this notion, for three classes of network systems, namely the ones with quadratic supply rates, with general nonlinear supply rates, and with linearly parametrized supply rates, small gain-like conditions for analysis, as well as feasibility conditions for control synthesis, are derived.We have then presented an algorithm to efficiently place the active nodes such that the feasibility condition is satisfied.In this case, the computation and adaptation of the damping coefficients of the active nodes are also discussed.To demonstrate how to apply the proposed scheme to a practical problem, a public health-related control problem has been discussed and simulation results have shown that by properly placing the active nodes and designing the quarantine forces of the active nodes via computation or adaptation, one can asymptotically "dissipate" the infectious population to zero without adding further quarantine measures to the noncritical nodes.
There are some issues that are not covered by the article.For example, in Section IV we have discussed the parametrization of nonlinear supply rates, but the scaling is not parametrized: This restricts computation of the nonlinear scaling.The reduction of the network system using active nodes is not discussed, which limits the use of recursive synthesis procedures for large-scale networks.These issues are under investigation and will be reported in future work.

Fig. 3 .
Fig. 3. Directed graph describing the migration among the 10 settlements.The green vertices denote the active nodes I A = {1, 2, 9} placed by Algorithm 3. The lower-right plot shows the remaining directed acyclic graph after deleting I A and the attached edges, verifying that I A is an FVS.

Fig. 4 .
Fig. 4. Time histories of the infectious/quarantined populations and the estimated quarantine forces in Scenarios 1-3.

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Recursive Algorithm to Compute the Lower Triangular Matrix M n T and the Pivot Gain Functions α and α (see Algorithm 2 for the Helper Functions).
[35], where k ≈ 1.264085, and • denotes the floor function.See[35]for detail.Algorithm 1 To linearize these nonlinear terms, define d j ĉi Aj ãi Aj , where i Aj ∈ ÎA is the vertex index of the jth (augmented) vertex that originates from the active nodes, for j = 1, . . ., | ÎA |, and d [d 1 , . . ., d | ÎA | ] .Note that the damping coefficients a i Aj can be obtained from d by using Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.