MINIMUM SENSITIVITY REALIZATIONS OF NETWORKS OF LINEAR SYSTEMS

. We investigate networks of linear control systems that are inter-connected by a ﬁxed network topology. A new class of sensitivity Gramians is introduced whose singular values measure the sensitivity of the network. We characterize the state space realizations of the interconnected node transfer functions such that the overall network has minimum sensitivity. We also develop an optimization approach to the sum of traces of the sensitivity Grami-ans that determine minimum sensitivity state space realizations of the network. Our work extends previous work by [6, 10, 11] on L 2 -minimum sensitivity de-sign.


1.
Introduction. Mathematical models for interconnected dynamical systems have for long played an important role in various application areas, including, e.g., biological systems, network control systems, flocking and formation control; see, e.g., [15,16,20] and the references therein. Typically, the components of a network are represented by linear dynamical systems, the so-called nodes, while the interconnections among the nodes define the network topology. The description of such networks depends on parameters defining node systems and the couplings between them. In such regard it is important to obtain a quantitative measure for the sensitivity of the interconnected system with respect to variations in the node systems and the interconnection parameters. This sensitivity measure for network of systems quantifies robustness properties of interconnected systems with respect to variations in the parameters. For a general approach to sensitivity in a transfer function context we refer to [1,2,25]. In the analysis of such coupled dynamical systems it is of fundamental interest to identify the state space representations of the node systems that contribute most, or least, to the sensitivity of the overall network. This problem is studied in the present paper.
Sensitivity issues of linear systems are important in digital controller and filter design, which are subject to finite precision errors and round-off noise [6,18,19]. For linear systems, there are several notions of sensitivity. A class of mixed L 1 /L 2sensitivity measures was studied by [18,19,22,23], and the associated minimumsensitivity realizations were shown to coincide with balanced realizations, first introduced by B.C. Moore [17]. Thus there exists an interesting connection between optimal sensitivity design and model reduction; see also [11]. A natural (and more complicated) L 2 -sensitivity measure has been studied by [6,11], with subsequent contributions by Helmke and Moore [10] and [13]. The corresponding sensitivity Gramians are however harder to compute and one has to recourse to numerical optimization methods, see [24].
In this paper, we extend this earlier work to interconnected linear systems, which poses two new problems. First, the known network structure has to be reflected in the solutions to the optimization tasks. Following [4], this is best done in the frequency domain, using coprime polynomial matrix factorizations of the node transfer functions. Second, the inherent structure of the parameter variations leeds to a structured optimization task.
We consider networked control systems consisting of N node systems Σ i , i = 1, ..., N , which are interconnected by constant coupling matrices. In particular, this framework includes the familiar processes of parallel, series and feedback interconnections. We assume that the interconnection structure of the network is described by fixed, constant matrices (K, L, M ), where K reflects the internal couplings of the node systems, while L and M describe the input and output couplings, respectively. The dynamics of each linear node system Σ i are described by strictly proper rational transfer functions G i , with minimal state space realizations (A i , B i , C i ). We introduce an L 2 -network sensitivity function that measures the performance degradation of the overall system with respect to variations, or parameter errors, in the realizations of the node system. To this end, we introduce a new class of sensitivity Gramians corresponding to the sensitivity function. Based on the equality of sensitivity Gramians we provide a complete characterization of sensitivity optimal realizations of the node systems. Computing minimum sensitivity realizations for the node systems can be done using optimization algorithms that minimize the sensitivity function. Gradient flows on positive definite matrices can be designed that converge to the minimum of the sensitivity function, similarly to [11]. As parallel, series and tree-like interconnections are the building blocks for connections of networks of linear systems we consider these structures and achieve explicit characterizations of minimum sensitivity realizations.
In this context it also reasonable to investigate the related question of sensitivity analysis with respect to variations in the interconnection matrices (K, L, M ), while keeping the node realizations (A i , B i , C i ) of the node systems fixed. This is not done here, and we refer to [9] for a sensitivity analysis of interconnection structures for networks of single-input-single-output systems.
The paper is organized as follows. Section 2 introduces the class of networks we are considering. In Section 3 we introduce a new class of Gramians, called the node sensitivity Gramians and discuss their relation to the classical Gramians. We derive necessary and sufficient conditions for positive definiteness of the Gramians. Section 4 introduces the sensitivity function and expresses it in terms of traces of the sensitivity Gramians. Furthermore, we investigate how the sensitivity Gramians transform under changes of coordinates in the node systems. Theorem 4.3 establishes the existence and uniqueness properties of sensitivity optimal realizations. Section 5 discusses computational issues. In Section 6 we apply the results to special interconnections, i.e., parallel and series interconnections of linear systems as well as networks with a tree structure. Conclusions appear in Section 7.
This work owes much to the early collaboration with John B. Moore. John introduced the first author to the sensitivity optimization problem of state space realizations of linear systems, a problem which is generalized here. Our joint research started in the late 1980s, when working on the related task of optimizing a trace function on positive definite matrices. That led us into studying optimization tasks on manifolds of matrices and resulted in several joint papers and the monograph [11]. The scientific collaboration and friendship with John, our discussions during daily walks at Lake Griffin, or on weekend excursions, have been special highlights of frequent visits at the Australian National University. I (U.H.) will never forget those moments.
2. Networks of Linear Systems. In this section we provide the conceptual framework to study the sensitivity of node transfer functions in a given network of linear control systems. Consider a network of N strictly proper linear discretetime systems Σ i : We refer to (1) as the node systems Σ i of the network. In the sequel, we focus on discrete-time systems; the equivalent statements for continuous-time systemṡ hold true mutatis mutandis. Here A i ∈ R ni×ni , B i ∈ R ni×mi and C i ∈ R pi×ni are the associated system matrices. We assume that each node system is reachable and observable and has the transfer function G i (z) = C i (zI − A i ) −1 B i . Without loss of generality we assume that B i , C i are full rank matrices, i.e., they satisfy rank(B i ) = m i and rank(C i ) = p i . To interconnect the node systems we use static coupling laws as with K ij ∈ R mi×pj and L i ∈ R mi×m , although more complex dynamic interconnection laws are possible, too. Here u = col(u 1 , ..., u m ) ∈ R m denotes the external control input applied to the whole network. The interconnected output is given by where n := n 1 + · · · + n N , m := m 1 + · · · + m N and p := p 1 + · · · + p N . To express the closed loop system in compact matrix form, we use for matrices X 1 , ..., X n the notation diag(X 1 , · · · , X n ) :=

UWE HELMKE AND MICHAEL SCHÖNLEIN
Then, for x(t) = col(x 1 (t), ..., x N (t)) ∈ R n and block-diagonal matrices, of sizes n × n, n × m and p × n, respectively, the global state space representation of the node systems is given by and the interconnection is given as Throughout this paper we call the matrix triple (K, L, M ) the interconnection matrices, while (A, B, C) are referred to as the node systems. Unless stated otherwise, the matrices (A, B, C) are always assumed to be in block-diagonal form (2). Thus the network dynamics has the state space description as Switching to a frequency domain description of the network we note that the i-th node transfer function is defined as the strictly proper transfer function of McMillan degree δ(G i ) = n i and is given in state space form We define the node transfer function as The overall network transfer function then is defined as We obtain the following representations of the global network transfer function It follows that the network transfer function F (z) is a function of the similarity orbit {(SAS −1 , SB, CS −1 | S ∈ GL n (R)} of a realization (A, B, C) of G(z). In the sequel, we aim to find block diagonal state space coordinate transformations S of the node systems such that the network transfer function F is least sensitive to variations in S. We emphasize that the network parameters K, L, M are just coupling parameters which are regarded as fixed.
3. Network Sensitivity Gramians. In real-world implementation of control systems, one is often concerned with the fact that different state-space realizations of the same given transfer function might perform differently. This raises the question of constructing state-space realizations with minimal sensitivity properties. In the literature several well-known sensitivity measures have been used; see e.g. [22,23]. For more extensive studies on minimum sensitivity design of linear systems we refer to [6,11]. With the exception of [9], the task of sensitivity minimization of networks has not been studied before. When studying sensitivity properties of networked control systems the earlier techniques have to be adapted appropriately. To obtain a measure to quantify the sensitivity of a networked control system with respect to parameterization errors we consider the global network transfer function F (z).
Node Sensitivity Gramians. To analyze the sensitivity of the network with respect to changes in the node system parameters A, B and C, we consider the Jacobians of the global network transfer function with respect to the block-diagonal matrices A, B, C. To this end, let More precisely, the matrices X and Y have the form where X i (z) ∈ R(z) ni×m and Y i (z) ∈ R(z) p×ni . Before turning to the Jacobians of the transfer function with respect to variations in (A, B, C) we emphasize that the matrices A, B and C are block diagonal. Recall that the Kronecker product of the matrices U ∈ R n×m and V ∈ R p×q is given by

Moreover, given block matrices
with A kl ∈ R n k ×n l and B kl ∈ R m k ×m l with k, l ∈ {1, ..., N } the block Kronecker product (sometimes also called the Khatri-Rao product) [12,14] is Lemma 3.1. The Jacobians of the global network transfer function F with respect to A, B, C are given by Proof. We begin with the Jacobian of the global network transfer function F with respect to B. For matrices X, Y, Z of appropriate size one has Using this identity it follows that This shows Finally, we consider the Jacobian of the global network transfer function F with respect to A. From the computation This shows the assertion.
We introduce a new class of Gramians Γ o , Γ c and Γ i of sizes These are symmetric and positive semidefinite matrices, which are called the input-, output-, and state interconnection sensitivity Gramians, respectively. Recall, that the classical reachability and observability Gramians associated with the network realization (A, B, C) are and The Gramians W c and W o exist and are positive definite, provided A = A + BKC is (discrete-time) stable, i.e. all eigenvalues are in the open unit disc, and (A, B, C) is reachable and observable. See [4] for a characterization for reachability and observability of (A, B, C) in terms of associated coprime factorizations; see also [8] for a reachability analysis of homogeneous networks, defined by identical SISO node systems. If (A, B, C) is reachable and observable, the Gramians W c and W o satisfy the discrete-time Lyapunov equations Define the n k × n k -matrices Thus W k o and W k c are the block-diagonal parts of W o and W o , respectively. By Lemma 3.1, the sensitivity Gramians and the classical Gramians are related as Define the local sensitivity Gramians for k = 1, . . . , N as Thus the local sensitivity Gramians are the block-diagonal parts of Γ c , Γ o and Γ i , respectively. The Cauchy-Schwarz inequality implies the bound on the state interconnection sensitivity Gramian where Tr[X] denotes the trace of the matrix X. In are positive definite. Note that the matrices L and M are typically tall and wide, respectively. For this reason one cannot expect that Γ c and Γ o are positive definite. Next, we derive sufficient conditions such that the sensitivity Gramians exist and are positive definite.
The same argument shows that Γ c is positive definite. The interconnection sensitivity Gramian Since (A, B, C) is reachable and observable with rank C = p and rank B = m there exist left and right coprime polynomial matrix factorizations with U (z) left prime and U (z) right prime. Thus there are polynomial matrices . By left and right primeness of U (z) and U (z) this implies V = 0. This shows the assertion. 4. Sensitivity optimal realizations. The L 2 -norm of a rational transfer function T (z) with no poles on the unit circle is defined as Following [11, Section 9.2] we define the following.
is called the weighted L 2 -network sensitivity of F (z) with respect to the node systems (A, B, C).
Using the definition of the sensitivity Gramians we obtain the following expression of the L 2 -network sensitivity. The proof is by a straightforward computation.
Proposition 2. Let G(z) be a strictly proper transfer function describing the decoupled node systems and let (K, L, M ) be the interconnection matrices. Then the weighted L 2 -network sensitivity is given in terms of the sensitivity Gramians as Dependence on Coordinate Transformations. We next discuss how the sensi- Let GL(n) = GL n1 (R) × · · · × GL n N (R) ⊂ GL n (R) denote the direct product of the N groups of invertible real matrices of sizes n 1 , . . . , n N . We regard the elements of GL(n) as block diagonal matrices of the form S = diag (S 1 , · · · , S N ),

UWE HELMKE AND MICHAEL SCHÖNLEIN
where S i ∈ GL ni (R). Thus GL(n) is a closed Lie group of GL n (R) of dimension n 2 1 + · · · + n 2 N . We now address the existence of sensitivity optimal realizations as well as their characterization. Since reachable and observable realizations of a strictly proper transfer function are similar, we restrict the sensitivity function to the orbits of the state space similarity action with S ∈ GL(n). Fix the node transfer function G as well as the interconnection parameters K, L, M . Choose a diagonal, reachable and observable realization (A, B, C) of G(z) with associated GL(n)-similarity orbit Then R G is a smooth manifold. Let Γ c , Γ o and Γ i denote the sensitivity Gramians of (SAS −1 , SB, CS −1 ). The L 2 -sensitivity function then defines the smooth function or, equivalently, the smooth function S → α Tr Γ c + β Tr Γ o + γ Tr Γ i on the space of block-diagonal invertible coordinate transformations. (a) The sensitivity Gramians transform as (c) The L 2 -network sensitivity function S G is unitarily invariant on R G .
Proof. To show (a), one verifies that Similarly The preceding lemma yields a convenient expression for the sensitivity function in terms of the positive definite matrix P = S T S as S G (P ) := α Tr Γ c (P ⊗ I) + β Tr Γ o I ⊗ P −1 + γ Tr Γ i (P ⊗ P −1 ) .
Characterization of Sensitivity Optimal Realizations. We next formulate our main result on the existence and uniqueness of L 2 -sensitivity optimal realizations.
Proof. By Proposition 1, the Gramians Γ c , Γ o , Γ i are positive definite. Obviously, we have S G (A, B, C) ≥ 0 for any (A, B, C) ∈ R G . For any S ∈ GL(n) it holds Note that α > 0 and β > 0. Moreover, by Lemma 6.4.1 in [11], for all c ∈ R, the set is a closed subset, the first claim follows. The second assertion follows immediately from the continuity of the L 2 -network sensitivity function.
The set R G of real similarity orbits is a closed submanifold of the complex manifold R C G , which is defined by all similarity orbits (SAS −1 , SB, CS −1 ) of complex invertible matrices S = diag(S 1 , · · · , S N ). By inspection, the L 2 -sensitivity extends to a plurisubharmonic function on R C G . Thus the proof of the claim follows as in [11], using the Azad-Loeb Theorem.
We next turn to analyze the critical points of the L 2 -sensitivity function. Let P := S T S and let P(n) denote the convex space of all real n × n block-diagonal, symmetric matrices P = diag(P 1 , · · · , P N ), where P i is n i × n i and positive definite. Similarly, let Sym(n) denote the vector space of all block-diagonal real symmetric matrices ξ = diag(ξ 1 , · · · , ξ N ), where ξ i has size n i × n i . Consider the function S G : P(n) → R defined by Note that the tangent space of P(n) at any element P is given by the vector space of real symmetric n × n-matrices Sym(n).
The differential DS G : Sym(n) → R is To compute the critical points of S G one has to find all P ∈ P(n) with DS G (P ) = 0. A computation shows The critical points of S G are thus characterized as the positive definite matrices P ∈ P(n) such that the block-diagonal parts P k satisfy for k = 1, . . . , N This implies the last claim.
One can improve Theorem 4.3 for special classes of networks. To obtain a simplified representation of the sensitivity Gramians, we focus on the case of a homogeneous network, where all node systems are identical SISO systems, represented as (A, b, c). Let g(z) = c(zI n − A) −1 b denote the transfer function with inverse h(z) = g(z) −1 . The state space representation of the network is then given as As is shown in [18] the following equalities hold The associated network transfer function is Recall that a scalar transfer function g(z) of a discrete-time system (A, b, c) is bounded real, if A is stable and |g(z)| ≤ 1 for all z with |z| = 1. A transfer function g(z) with all poles inside the open unit disc is called lossless if |g(z)| = 1 for all z with |z| = 1. By the bounded real lemma, g(z) is bounded real if and only if A is stable and there exists a positive definite solution of the algebraic Riccati equation For symmetric matrices A, B, let A ≤ B denote that B − A is positive semidefinite.
In [13,Lemma 5] it is shown that, if the transfer function g is bounded real, then for the controllability and observability Gramians for (A, B, C) and (K, L, M ) the following inequalities hold

UWE HELMKE AND MICHAEL SCHÖNLEIN
where P is the unique positive definite solution to the Riccati equation (9). Moreover, if g(z) is lossless, then Note that, if (A, b, c) is replaced by (SAS −1 , Sb, cS −1 ), the unique solution of (9) is given by S −T P * S −1 . Thus, for g bounded real and P * defined as above, one has for k = 1, . . . , N If g(z) is lossless, then equalities in (11) hold. Next assume that α = β = 1, γ = 0. Then the input/output sensitivity function is given as Theorem 4.4. Let g(z) = c(zI − A) −1 b be bounded real and let P * be a solution of the Riccati equation (9). Define Then the minimum value min S co over all realizations of g(z) satisfies the bound min S co ≤ 2n λµ.
If g(z) is lossless, then min S co = 2n λµ, with an optimal state space coordinate transformation S * = µλ −1 P 1 2 * . Proof. By (11), for all P = SS T = λ Tr P −1 * P + µ Tr P −1 P * It is easily seen that the smooth function Q → λ Tr Q + µ Tr Q −1 has a unique local and global minimum at Q min = µλ −1 I n . Thus the optimal transformation P min minimizing λ Tr Q + µ Tr Q −1 is equal to µλ −1 P * . This implies the result.
The upper bound in Theorem 4.4 depends on the fact that we work with homogeneous networks of identical SISO systems. This allows one to prove the important inequalities (10). We leave it as an open problem to find extensions for heterogeneous networks of SISO node systems.
We briefly discuss how the preceding computations simplify for transfer functions with symmetries. Recall that a realization (A, B, C) is called signature symmetric provided there exists a diagonal matrix J with diagonal entries ±1 such that (AJ) T = AJ and CJ = B T holds. J is called the signature matrix and satisfies J = J −1 . By a well known result from linear systems theory, any strictly proper rational transfer function G(z) which satisfies G(z) = G(z) T has a signature symmetric realization. Moreover, the difference in the numbers of positive and negative diagonal entries in J coincides with the matrix Cauchy-index of G(z).
In particular, a signature symmetric realization (A, B, C) with signature matrix J = I has minimal L 2 -sensitivity.
Proof. We have Y (z) = JX(z) T and thus the result follows from part (c) of Theorem 4.3.
5. Computational Issues. We briefly address some computational issues, focusing on input and output sensitivities (i.e., we set γ = 0).
Computation of Gramians. By inspection of the contour integrals that define the sensitivity Gramians (3), (4) and (5) one might wonder how to effectively compute them. One approach is to use numerical methods for computing contour integrals. Let ω = exp( 2πiτ s ) denote a primitive s-th root of unity we consider the k, l-entries of the approximate Gramians, being defined as Choose any 0 < r < 1 < 1 r such that A + BKC has all its eigenvalues outside the open annulus A(r) = {z ∈ C : 0 < r < |z| < 1 r } and not on |z| = 1 r . Then, using a result by [3], there is a κ > 0 the such that the approximation error Γ o − Γ s o satisfies the following bound

UWE HELMKE AND MICHAEL SCHÖNLEIN
Similar estimates hold for the other two Gramians. Note that r can be chosen arbitrarily small, provided A + BKC is nilpotent.
A second approach utilizes the block-Hadamard product representation (7) and computes the Gramians W c and W o , by solving the discrete-time Lyapunov equations (6). A difficulty with this approach, for N large, lies in the large-scale nature of the Lyapunov equations The network structure of (A, B, C) cries for a distributed algorithm to solve (6). This will be done elsewhere.
Frequency Gramians. If all node transfer functions are SISO and identical, then frequency domain representations of the sensitivity Gramians can be obtained, similarly to [9]. Thus assume that all node systems are identical SISO systems, and are given by (A, b,  Computing Sensitivity Optimal Realizations. Numerical algorithms for computing sensitivity optimal realizations have been proposed by [11], using either gradient steepest descent schemes or discrete-time Riccati equations. Here we propose a simple Newton algorithm that is fast (locally quadratically convergent) and globally convergent. For simplicity, we assume α = β = 1 and γ = 0, i.e., we focus on the task of minimizing the sensitivity function defined by The analysis can be extended to the general sensitivity function Tr Γ c + Γ o + Γ i , but at the expense of dealing with complicated expressions for the Newton step.
Let (A, B, C) be reachable and observable. Define the block-diagonal matrices To optimize the cost function S co we first consider the simplified task of optimizing the cost function on the space P(n) of block-diagonal positive definite matrices P = diag (P 1 , · · · , P N ) as S co (P ) = Tr Γ c (P ⊗ I) The gradient of (12), with respect to the Riemannian metric ξ, η P := Tr P −1 ξP −1 η , ξ, η ∈ T P P(n) on P(n), is easily computed as grad S co (P ) = diag (∇ 1 , · · · , ∇ N ) . . , N. To simplify the expressions for the subsequent formulas we assume that M k = L k = 1 for all k. Then the gradient becomes grad S co (P ) = P W c P − W 0 .
For P ∈ P(n), let Z = Z(P ) ∈ P(n) denote the unique positive definite solution of the Lyapunov equation Since W c , W o and P are block diagonal, the solution Z(P ) = diag (Z 1 , · · · , Z N ) is unique and block-diagonal. Thus solving (13) is equivalent to solve the decoupled system of Lyapunov equations We use the notation P > 0 to denote that P is symmetric and positive definite. For any initial point P 0 ∈ P(n) with P 0 W c P 0 − W o > 0, we consider the iterative system P t+1 = P t − Z(P t ), t = 0, 1, . . . .
Note that ZW c P + P W c Z is just the directional derivative of the map P → P W c P − W o in direction Z. Thus (14) is a Newton algorithm for the strictly convex cost function S co on P(n). Thus we expect global quadratic convergence of the algorithm. This is confirmed by the following theorem.
Theorem 5.1. Let W c , W o denote the block diagonal parts of the sensitivity Gramians Γ c , Γ o of a reachable and observable realization (A, B, C). Assume P 0 ∈ P(n) is chosen such that P 0 W c P 0 − W o is positive definite. Then the Newton iterates (14) converge monotonically and locally quadratically fast to the sensitivity optimal P * ∈ P(n). More precisely, we have (a) (P t ) t is monotonically decreasing , i.e. P t > P t+1 holds, and converges to the unique positive definite P * satisfying P * W c P * = W o . (b) (P t ) t converges locally quadratically fast to P * . A sensitivity optimal coordinate transformation S * ∈ GL(n) is obtained by computing a Cholesky factorization S T * S * = P * of P * . Proof. The symmetry of P t is obvious. Assume P t is positive definite. Then P t+1 is a solution of the Lyapunov equation with P t W c having all eigenvalues with positive real part. Thus P t+1 is positive definite. We compute Monotonicity follows from P t W c P t > 0 and P t W c Z t + Z t W c P t > 0 and therefore P t − P t+1 = Z t > 0. Thus the limit P * = lim t→∞ P t exists. From the iteration we obtain Z(P * ) = 0 and therefore P * W c P * = W o holds. Finally, local quadratic convergence follows, since the iteration is a Newton algorithm and S co is strictly convex.
6. Input-output sensitivity balancing of standard interconnections. The building blocks of interconnected systems are given by parallel, series and tree-like interconnections. To this end, we apply in this section the results of Section 4 to linear systems connected in parallel, series and via an interconnection graph being a tree.
Parallel connection. The parallel connection of N realizations (A i , B i , C i ) is defined by the interconnection matrices K = 0, L = col(I m , · · · , I m ) and M = (I p , · · · , I p ). Thus, the realization (A, B, C) of the networked control systems is given by We assume that the parallel connected system (A, B, C) is reachable and observable. For necessary and sufficient conditions of reachability of parallel connected systems we refer to [4,5]. The block-diagonal parts of the sensitivity Gramians Γ c and Γ o are diag(W (1) c , · · · , W (N ) denote the standard discrete-time reachability and observability Gramians of (A k , B k , C k ).  Proof. The first assertion follows immediately from Theorem 4.3, while the other assertions follow from Corollary 1.
Series connection. Analyzing sensitivity optimization for series connections of systems is more difficult. We focus on input and output sensitivities. The interconnection matrices for a series connection of N realizations (A k , B k , C k ) are

The realization (A, B, C) of the series connection is
To determine the input/output sensitivity Gramians we define G N k (z) := G N (z) · · · G k (z), k = 1, . . . , N. We compute . . .
for l = 1, ...., N −1. Here W c (A k , B k ) and W o (C k , A k ) denote the standard discretetime reachability and observability Gramians of (A k , B k , C k ), respectively. Recall that a transfer function G is allpass, if G(z)G(z −1 ) T = I holds for all |z| = 1. The preceding computation shows. Tree interconnections. Coupling N state space systems (A k , B k , C k ) in form of a tree, leads, after a suitable permutation, to interconnection matrices Here K ij is zero, whenever the node system i is not connected with node j, and L i = I if the node i has a nonzero in-degree, L i = 0 otherwise. Similarily, M i = I if node i has a nonzero out-degree, M i = 0 otherwise. For simplicity, we consider the example of three systems in series/parallel connection, see Figure 1. A calculation yields Consequently, the diagonal entries of the sensitivity Gramians are Consequently, we obtain the following Theorem 6.3. Assume that the transfer functions G 1 (z) and G 2 (z) + G 3 (z) are allpass. Then the series/parallel connection (A, B, C) is sensitivity minimal if and only if (A k , B k , C k ) are balanced realizations for k = 1, 2, 3.

7.
Conclusions. In this paper we consider heterogeneous networks of N linear systems, where the node dynamics are given by multi-input-multi-output (MIMO) linear systems Σ i , represented by strictly proper transfer functions G i (z). The interconnection structure of the networked system is regarded as fixed and known. We extend the sensitivity analysis from single systems to interconnected systems where we considered the sensitivity of the network transfer function with respect to the realizations (A, B, C) of the node systems. To this end, we adapt the wellknown L 2 -sensitivity measure for single linear systems to the class of heterogeneous networks of linear MIMO systems. A new class of network sensitivity Gramians is introduced whose traces provide the sensitivity measure. Existence and uniqueness of sensitivity optimal realizations is shown. Also we discuss the linear systems connected in series and in parallel as well as networks with tree structure.