Large-Scale Dissipative and Passive Control Systems and the Role of Star and Cyclic Symmetries Vahideh Ghanbari, Po Wu, and Panos J

Home , Dissipative system

Large-Scale Dissipative and Passive Control Systems and the Role of Star and Cyclic Symmetries Vahideh Ghanbari, Po Wu, and Panos J. Antsaklis

Abstract—In this paper, symmetries in large-scale dissi- the entire class of systems was determined by reducing pative and passive control systems are considered. In the the model and examining a lower order equivalent class. framework of dissipativity and passivity theory, stability Different forms of symmetry, such as star-shaped or cyclic conditions for large-scale systems are derived by categorizing agents into symmetry groups and applying local control symmetry result in distinct stability conditions for inter- laws under limited interconnections with neighbors. Building connected systems. We will also show this in the present upon previous studies on stability of (Q, S, R) − dissipative paper using dissipativity theory. large-scale systems, we show that for cyclic and star-shaped The notion of dissipativity is a generalization of Lya- symmetric systems there exists an upper bound on the punov stability theory where an energy like quantity associ- number of subsystems that can be added to preserve stability of dissipative systems. In cyclic and star-shaped symmetric ated with a system is defined to be non-increasing over time systems, the subsystems can be heterogeneous as long as [10]. A dissipative system can store or dissipate generalized they satisfy the same dissipative inequalities. Approximate energy supplied to the system without generating surplus symmetry with respect to interconnections is also considered energy. The energy is characterized by a storage function, and the robustness of the results is demonstrated. which is a generalization of a Lyapunov function. For dissipative systems, the energy can increase, as long as the rate of increase is bounded by a supply rate, which can be a I.INTRODUCTION function of input, output, state and time. Electrical circuits Symmetry is a basic feature of shapes and graphs and and other physical systems are common examples [10]. can be found in many real-world networks, such as the Nevertheless, the notion of a generalized energy storage Internet and power grids, as a result of tree-like or cyclic function, supply rate, and dissipativity can be applied to growth process. Since symmetry is related to the concept general nonlinear systems. Passivity is a special case of dis- of high degree of repetitions or regularities, common in sipativity, where the supply rate has a special form. When nature, the study of symmetry has been undertaken in many two passive systems are interconnected in parallel or in scientific areas, such as mathematics (Lie groups), quantum feedback configuration, the overall system remains passive. mechanics, and crystallography in Chemistry. Thus, passivity may be preserved when large-scale systems The concept of symmetry has been studied in the are put together from components of passive subsystems, classical theory of dynamical systems. For example, to which makes the passivity an appealing property in large- simplify the analysis and synthesis of large-scale dynamical scale systems. systems, it is of interest to consider smaller symmetric sub- Dissipativity and passivity in systems, together with systems with simplified dynamics, which may potentially decomposition into lower order subsystems have been simplify the analysis in control, planning or estimation [1], previously used in the study of large-scale systems. Lya- [2]. When dealing with multi-agent systems with various punov stability has been studied using vector Lyapunov information constraints and communication protocols, un- functions, and input-output stability results have been pre- der certain conditions such systems can be expressed as sented for dissipative subsystems [4]. In addition, Ref. or decomposed into interconnections of lower dimensional [11] studied interconnections of dissipative subsystems as systems, which may lead to better understanding of system well as special types of dissipative systems such as finite properties such as stability and controllability [3], [4]. gain systems, and passive systems. Ref. [12] generalizes Early research on symmetry in dynamical systems can previous results to weighted Lyapunov functions and gives be found in [5]–[7]. Symmetry in the context of distributed spectral characterization of the interconnection matrix. The systems containing multiple instances of identical subsys- stability conditions in interconnected passive systems, and tems are studied in [8], [9], where the controllability of the close relationships between output feedback passivity and L2 stability have been studied in [13], [14]. Recently, The authors are with the Department of Electrical Engineering, at the University of Notre Dame, Notre Dame, IN 46556 USA (email: the use of diagonal stability in study of passive network [email protected]; [email protected]; [email protected]). systems have been investigated [15]. Moreover, a passivity- based controller for asymptotic stabilization of interconnected port-Hamiltonian systems has been designed [16]. _ In addition, the study of consensus algorithms for multi- agent network systems based on the graph Laplacian of the networks is represented in [17]. . . In the present paper, first (Q, S, R) − dissipativity + .

of large-scale systems are studied as an extension of . . the results in [11]. Next, stability conditions for large- . scale systems consisting dissipative subsystems are derived by categorizing agents into symmetry groups and applying local control laws under limited interconnections Figure 1. Interconnected multi-agent system. Hi1, ..., HiN are constant interconnection matrices and H is constant local feedback matrix. with neighbors. Conditions are derived for the maximum i number of subsystems that can be added while preserving stability and these results may be used in the synthesis matrix. of large-scale systems with symmetric interconnections. Let Q = −I, S = 0 and R = k2I, for some fixed positive This paper mainly focuses on two kinds of symmetry: star- real number k. In this case it can be shown that the equation shaped symmetry and cyclic symmetry. The present work (3) implies extends the previous study [18] to additional symmetry kykT ≤ k kukT , (4) cases and presents new results in the case of (Q, S, R) − 2 where k k is the truncated norm, defined via kxk = dissipativity, passivity, and approximate symmetries. T T hx, xi . If (4) is satisfied, we say that the system is finite The paper is organized as follows: In Section II, we T gain input-output stable, or L2 −stable with an upper gain introduce a background on dissipativity, passivity and sym- bound of k. metry in dynamical systems. Section III describes stability Consider N dissipative subsystems and each subsystem conditions for dissipative and passive systems with star- Σ in Figure 1 has the dynamics shaped symmetry interconnections. Section IV discusses stability conditions for dissipative systems with cyclic sym- x˙ i = fi(xi) + gi(xi)ui, yi = hi(xi), (5) metry, and finally, the concluding remarks are presented. that satisfies the dissipativity inequality with storage func- T T tion Vi(x) and supply rate ωi(ui, yi) = yi Qyi +2yi Sui + II.BACKGROUNDAND PRELIMINARIES T ui Rui, where ui, yi are input/output with the same dimen- Consider a nonlinear system sion (m = p). Let the linear feedback interconnections be described by x˙(t) = f(x(t)) + g(x(t))u(t), y(t) = h(x(t)), (1) N X where x(t) ∈ X ⊂ Rn, y(t) ∈ Y ⊂ Rp, u(t) ∈ U ⊂ Rm, ui = uei − Hijyj, i = 1, ··· ,N, (6) and m ≤ n. f(.) ∈ Rn, g(.) ∈ Rn×m and h(.) ∈ Rp are j=1 continuous functions where f(0) = 0, h(0) = 0 and f(.) where ui is the input to the subsystem i, yi is its is a locally Lipschitz function. Let U be an inner product output, uei is an external input, and Hij are constant space whose elements are functions f : R → R. Also let matrices. The system inputs and outputs are stacked as m T T T T T T T T U be the space of m-tuples over U, with inner product u = [u1 , u2 , . . . , u ] , y = [y1 , y2 , . . . , y ] . Defining Pm N N hy, ui = i=1 hyi, uii. a constant interconnection matrix, H = [Hij], the linear interconnected system can be represented by Definition 1. [10] A dynamical system with supply rate ω(u, y) is dissipative if there exists a positive definite u = ue − Hy. (7) storage function V (x) such that for all t < t 1 2 Lemma 1. [11] Let the ith subsystem have finite gain t2 Z input-output stable with gain γi, for i = 1, ··· ,N, and ω(u, y)dt ≥ V (x(t2)) − V (x(t1)). (2) suppose that each subsystem has only one input and one t1 output. Define Γ = diag {γ1, ··· , γN } and A = ΓH, Definition 2. [11] A system is (Q, S, R) − dissipative if where H can be obtained from (7). Then, if there exists a the system is dissipative with respect to the supply rate diagonal positive definite matrix D such that T y QS y T ω(u, y) = (3) D − A DA > 0, (8) u ST R u the interconnected system is BIBO stable. where Q ∈ Rp×p, S ∈ Rp×m, and R ∈ Rm×m. A sufficient condition for the existence of a matrix D, which satisfies (8) is that the matrix Passivity is a special case of (Q, S, R) − dissipativity, 1 ˆ i.e. Q = 0,S = 2 I, and R = 0, where I is the identity A = [âij] (9) has positive leading principal minors [19], where aîi = (SISO), i.e. m = p = 1. 1 − |aii|, aîj = − |aij| , i 6= j with A = [aij]. Note that Consider the nonlinear system (1) with the linear feed- Aˆ is positive definite in this case. back interconnections (7). The base system (y0, u0) is the starting system, which we expand about by adding Lemma 2. [12] If there exists a diagonal matrix D > 0 subsystems; (y , u ) i = 1,...,N denote the additional such that the matrix i i subsystems to the base system. For star-shaped symmetry, Qˆ = −HT DRH + DSH + HT ST D − DQ (10) we have the matrix of interconnections, H, in the form of   is positive definite, i.e. Qˆ > 0, then the network of N h0 h12 ...... h12 interconnected (Q ,S ,R )-dissipative agents (5), (6) is  h21 h0 0 ... 0  i i i   asymptotically stable.  . .. .  H =  . 0 . .  , (11)   Note that the existence of a diagonal matrix D > 0 is  . . ..   . . . 0  required in Lemmas 1 and 2, which links our approach to h 0 ... 0 h the diagonal stability analysis [15]. 21 0 In this paper, we consider two types of symmetries, where block entries of H are feedback scalar gains since namely star-shaped symmetry and cyclic symmetry. In a we consider each subsystem to be single-input single- star-shaped symmetry, subsystems do not have intercon- output. Thus, the linear feedback interconnections are PN nections with each other but the base system has inter- described by u0 = ue0 − h0y0 − i=1 h12yi, ui = connections with all the subsystems. In cyclic symmetry, uei − h21y0 − h0yi, i = 1,...,N. subsystems have interconnections with their neighbors and Theorem 1. Consider a finite gain system extended by N the base system (See Figures 2 and 3.) star-shaped finite gain symmetric subsystems with symmetric interconnections matrix H. If σ(Aˆ) N < , (12) −1 Subsystem σ(α12Aˆ α21) where Aˆ is defined in (9) and σ and σ are its minimum and Base maximum eigenvalues respectively, then the interconnected System system is BIBO stable. Proof. When the interconnected system is extended with ˆ hˆ i one subsystem, we have the new matrix Ae = eaij , where ˆ ˆ eaii = 1 − |eaii| , eaij = − |eaij| , i 6= j. eaij are the entries of the matrix Figure 2. Star-shaped symmetry γ0 0 h0 h12 Ae = ΓeHe = diag( ) . 0 γ1 h21 h0 ˆ ˆ Aˆ α12 The new matrix Ae can be written as Ae = , α21 Aˆ Subsystem where α12 = γ0h12 and α21 = γ1h21. Since Aˆ already has ˆ positive leading principal minors, Ae is an M-matrix [19] Base ˆ ˆ−1 System if and only if A − α12A α21 > 0 (Schur’s theorem, [20, p. 7]). Then, based on Lemma 1, the extended system is BIBO stable. Similarly, if the system is extended with N symmetric subsystems, we have Ae = ΓeHe and   Aˆ α12 ...... α12 ˆ Figure 3. Cyclic symmetry  α21 A 0 ... 0    ˆ  . . .  Ae =  . 0 .. .  III.DISSIPATIVE AND PASSIVE STAR-SHAPED    . . ..  SYMMETRIC SYSTEMS  . . . 0  ˆ A. Dissipative System with Star-shaped Symmetric Dissi- α21 0 ... 0 A pative Subsystems −1 is an M-matrix if and only if Aˆ − Nα12Aˆ α21 > 0, i.e. In this section, we use the results from Lemma 1 and there is an upper-bound on such symmetric extension given consider all the subsystems as single-input single-output by (12). Remark 1. The existence of such upper bound implies gains of subsystems connected to the base system in a that in order to preserve stability, positive feedback for star-shaped symmetric configuration. Then we have V˙ ≤ T T T networked control systems should be restricted. Since we y Qy+2y S[ue −Hy]+[ue − Hy] R[ue −Hy] and after assume feedback from symmetric subsystems, their feed- a few lines of algebraic calculations, we get V˙ ≤ yT [Q − T T T T back gains should be identical, either all negative feedback 2SH + H RH]y + y [2S − H R − RH]ue + ue Rue. or positive feedback. For the former case, the system Thus, the enlarged system is (Q,ˆ S,ˆ Rˆ) − dissipative with will remain stable even after adding arbitrary number of Qˆ, Sˆ, and Rˆ given by (14). subsystems. The result for negative feedback is a well Corollary 1. (Q, S, R) − dissipative known result [10]. For the positive feedback case, studied Consider a base N (q, s, r) − here, there exist an upper bound on the gain if stability system extended by star-shaped symmetric dissipative Σ is to be guaranteed, or an upper bound on the number of subsystems (Figure 2). System s is asymp- subsystems that can be added. Theorem 1 states that even totically stable if with very small gain (Lemma 1), stability may be lost for σ(Qˆ) qˆ N < min( , ), large numbers of subsystems. T −1 T (15) σ(H21rH21 + β(ˆq − H) β ) H For more general case, with appropriate change of nota- where tion to Kronecker algebra, we can get the results for multi- ˆ T T T input multi-output (MIMO) systems [12]. We consider Q = −H0 RH0 + SH0 + H0 S − Q > 0, (q, s, r) − dissipative subsystems connected to the base qˆ = −HT rH + sH + HT sT − q > 0, system in star-shaped symmetry. 1 1 1 1 Consider MIMO subsystems with dynamics shown in T T T T β = SH12 + H21s − H0 RH12 − H21rH1, (5) and linear interconnected system (7). All the (q, s, r)− T dissipative subsystems are connected to the base system H = H12RH12. in star-shaped symmetry and it is assumed that all sub- Proof. In Theorem 2, we have shown that the enlarged systems satisfy the dissipative inequalities with the same system is (Q,ˆ S,ˆ Rˆ) − dissipative. For simplicity, we (q, s, r). The constant interconnection matrix is consider D = I in Lemma 2 such that (10) holds. Based   H0 H12 ...... H12 on the linear interconnections, we have matrix of feedback  H21 H1 0 ... 0  interconnections, H, for star-shaped symmetry and for    . .. .  asymptotic stability we require H =  . 0 . .  (13)    . . ..  Qˆ = SH + HT ST − HT RH − Q > 0. (16)  . . . 0  H21 0 ... 0 H1 This implies and therefore, the linear feedback interconnections can be  T  N Q − NH21rH21 β β . . . β described by u = u − H y − P H y , u = T 0 e0 0 0 i=1 12 i i  β  u −H y −H y , i = 1,...,N where H and H , are  T  ei 21 0 1 i 12 21 Qˆ =  β Λ  > 0, (17) constant matrices containing feedback gains between the    .  base system and subsystems. Also, H0 and H1 are constant  .  local feedback matrices of base system and subsystems βT respectively. We now have the following result: where Theorem 2. Consider a (Q, S, R) − dissipative base β = SH + HT sT − HT RH − HT rH , system extended by N star-shaped symmetric (q, s, r) − 12 21 0 12 21 1 T dissipative subsystems as in Figure 2. System Σs is Λ =q ˆ ⊗ IN − H12RH12 ⊗ circ([11 ··· 1]). (Q,ˆ S,ˆ Rˆ) − dissipative with AB T According to the Schur’s theorem, > 0 if and Qˆ = Q − 2SH + H RH BT D 1 only if D > 0 and A − BD−1BT > 0. Thus, there is an Sˆ = S − {RH + HT R} (14) 2 upper-bound on such symmetric extension which is given Rˆ = R by (15). Proof. The base system is (Q, S, R) − dissipative with Remark 2. The right hand side of (15) consists of two supply rate ω(u, y) = yT Qy+2yT Su+uT Ru. Thus, there parts, the first part implies that the base system will become exists a storage function V > 0 such that V˙ ≤ ω(u, y). unstable if the sum of a positive feedback gains from all Also, the subsystems are added to the base system with subsystems exceed certain value, while the second part the linear feedback interconnections u = ue − Hy, where implies that the subsystem will become unstable if the ue is the external input (Figure 1) and matrix H contains positive feedback from the base system is too strong. Notice that hitherto we have considered all subsystems Consider a dissipative system extended by N star-shaped being (q, s, r) − dissipative. No restrictions were placed approximately symmetric dissipative subsystems. System on the actual dynamics which may be different from Σs is asymptotically stable if Qˆ > 0 in (10). The proof is each other. Thus, the above results apply to heterogeneous similar to that of Corollary 1. systems as well, as long as they satisfy the inequality (2). Example 1. Suppose we have N + 1 symmetric passive subsystems in Figure 2, and the linear feedback inter- B. Passive System with Star-shaped Symmetric Passive connections are described by u0 = ue0 − y0 + 0.4y1 + Subsystems 0.2y2 + ... + 0.2yN , u1 = ue1 + 0.4y0 − y1, ui = Corollary 2. Consider a passive system extended by N uei +0.2y0 −yi, i = 2,...,N. In this example, we consider star-shaped symmetric passive subsystems (Figure 2). Sys- SISO subsystems with scalar feedback gains. According to Corollary 3 (19), N < 6.25. Thus Nmax = 6, which is tem Σs is asymptotically stable if relatively conservative as an upper bound of the number σ(Qˆ) of subsystems that can be added to the base system in N < , (18) σ(βqˆ−1βT ) symmetric interconnections without losing stability.

T T ˆ H0+H0 H1+H1 where Q = 2 > 0, qˆ = 2 > 0, and β = T IV. DISSIPATIVE AND PASSIVE CYCLIC SYMMETRIC H12+H21 2 . The proof is similar to the proof of Corollary 1 SYSTEMS and is thus omitted here. A. Dissipative System with Cyclic Symmetric Dissipative Note that passive systems are special cases of dissipative Subsystems systems where Q = 0,S = 1 I,R = 0 for the base system 2 For cyclic symmetry, the matrix of linear interconnec- and q = 0, s = 1 I, r = 0 for subsystems. One can directly 2 tions, H is represented by get this result from equation (15) in Corollary 1.   H0 H12 ...H12  H21  C. Approximate Star-shaped Symmetry in Interconnected H   (20) ,  .  Passive and Dissipative Systems  . He  Exact symmetry in control systems may not apply in H21 some cases, for instance the models of distributed systems where H = circ([v v ··· v ]) is a circulant matrix may be different but bounded by some induced matrix e 0 1 N−1 with ﬁrst row [v v ··· v ]. Here H = P T HP and H norm, interconnections may have different weights, there 0 1 N−1 e e e H = v I + v P + ··· v P N−1.P may be time-variant delays, packet drops, etc. Disturbances can be written as e 0 1 N−1 is [1, 0, ..., 0] in system dynamics and interconnection structure can rep- a matrix in companion form with the last row . resent another type of approximate symmetry. The study We now consider subsystems with dynamics (5) and of approximate symmetry can provide more robust results linear interconnected system (7) with (q, s, r)−dissipative for dynamical systems. There are also a few results of subsystems, connected via a cyclic interconnection as approximate symmetry and approximate model reduction shown in Figure 3. of dynamic systems [21]. Corollary 4. Consider a (Q, S, R) − dissipative system When introducing approximate symmetry into dissipa- extended by N cyclic symmetric (q, s, r) − dissipative tive systems, the linear feedback interconnections can be subsystems as in Figure 3. System Σc is asymptotically PN described by u0 = ue0 − H0y0 − j=1 Hijyi, ui = stable if uei − Hjiy0 − Hiyi, i = 1,...,N. Therefore, star-shaped σ(Qˆ) qˆ subsystems have different constant matrices, although the N < min( , ), T −1 T T (21) structure of the network remains star-shaped symmetric. σ(H21rH21 + βN Λ βN ) H12RH12 Corollary 3. Consider a passive system extended by N where star-shaped approximately symmetric passive subsystems T T Λ = −H1 rH1+sH1 + H1 s − q ⊗ IN (Figure 2). System Σs is asymptotically stable if T −H12RH12 ⊗ circ([11 ··· 1]), σ(Qˆ) N < , i = 1, 2,...,N, (19) −1 T ˜ ˜ ˜ ˜ maxσ(βiqˆi βi ) qˆ = −rσ(H)σ(H) + s(σ(H) + σ(H)) − q, i

T T ˆ H0+H0 Hi+Hi T T T T where Q = 2 > 0, qˆi = 2 > 0, and βi = β = SH12 + H21s − H0 RH12 − H21rH,e T H12+H21 2 . The proof is similar to Corollary 1 and is βN = [ββ ··· β], therefore omitted here. | {z } N N−1 N−1 X j X 2πij REFERENCES N σ(He) = vjλi = vje . j=0 j=0 [1] J. Cortes, S. Martinez, J. P. Ostrowski, and H. Zhang, “Simple mechanical control systems with constraints and symmetry,” SIAM Remark 3. Similar to (15), the right hand side of (21) Journal on Control and Optimization, vol. 41, no. 3, pp. 851–874, consists of two parts, indicating that the positive feedback 2002. [2] Y. Wang and M. Morari, “Structure of hierarchical linear systems between the base system and subsystems should not be too with cyclic symmetry,” Systems and Control Letters, vol. 58, no. 4, strong. However, unlike the star-shaped structure, the ring- pp. 241–247, 2009. shaped structure does not necessarily need the support of [3] D. D. Siljak, Large-scale dynamic systems: stability and structure. North-Holland New York, 1978, vol. 310. the base system, which means no constraints are put on N [4] A. N. Michel, “On the status of stability of interconnected systems,” as in (21) if there is no base system [12]. In this case, H0 , Automatic Control, IEEE Transactions on, vol. 28, no. 6, pp. 639– q 2 653, 1983. H˜ = circ(v) σ(H˜ ) − s < s − q , and we have r r2 r where [5] F. Fagnani and J. C. Willems, “Representations of symmetric linear

PN−1 j PN−1 j PN−1 dynamical systems,” SIAM Journal on Control and Optimization, σ(He) = j=0 vjλi ≤ j=0 vjλi ≤ j=0 kvjk. vol. 31, no. 5, pp. 1267–1293, 1993. [6] J. Grizzle and S. Marcus, “The structure of nonlinear control systems Here, σ(He) is bounded and the bound will not be affected possessing symmetries,” IEEE Transactions on Automatic Control, by the number of subsystems given a similar structure of vol. 30, no. 3, pp. 248–258, 1985. the interconnection matrix, in which zero entries are ﬁlled [7] S. Y. Zhang, “The structures of symmetry and similarity of complex control systems,” Control Theory and Applications, vol. 11, no. 2, when adding new symmetric subsystems. pp. 231–237, 1994. [8] M. B. McMickell and B. Goodwine, “Reduction and nonlinear B. Passive System with Cyclic Symmetric Passive Subsys- controllability of symmetric distributed robotic systems with drift,” IEEE International Conference on Robotics and Automation, vol. 4, tems pp. 3454–3460, 2002. Corollary 5. Consider a passive system extended by N [9] ——, “Motion planning for nonlinear symmetric distributed robotic formations,” The International Journal of Robotics Research, cyclic symmetric passive subsystems (Figure 3). System vol. 26, no. 10, pp. 1025–1041, 2007. Σc is asymptotically stable if [10] J. C. Willems, “Dissipative dynamical systems part i: General theory,” Archive for Rational Mechanics and Analysis, vol. 45, no. 5, σ(Qˆ) pp. 321–351, 1972. N < , −1 T (22) [11] P. Moylan and D. Hill, “Stability criteria for large-scale systems,” σ(βN Λ βN ) IEEE Transactions on Automatic Control, vol. 23, no. 2, pp. 143– where 149, 1978. [12] S. Hirche and S. Hara, “Stabilizing interconnection characterization T T H1 + H1 H12 + H21 for multi-agent systems with dissipative properties,” Proceedings of Λ = , β = , βN = [ββ ··· β]. the 17th IFAC World Congress, pp. 1571–1577, 2008. 2 2 | {z } [13] M. Arcak and E. D. Sontag, “A passivity-based stability criterion for N a class of interconnected systems and applications to biochemical This Corollary can be directly concluded from Corollary reaction networks,” 46th IEEE Conference on Decision and Control, 1 pp. 4477–4482, 2007. 4 by putting Q = 0,S = 2 I,R = 0 for the base system 1 [14] E. D. Sontag and M. Arcak, “Passivity-based stability of intercon- and q = 0, s = 2 I, r = 0 for subsystems. nection structures,” Recent Advances in Learning and Control, vol. 371, pp. 195–204, 2008. [15] M. Arcak, “Diagonal stability on cactus graphs and application to V. CONCLUSION network stability analysis,” Automatic Control, IEEE Transactions In this paper, (Q, S, R) − dissipativity and stability on, vol. 56, no. 12, pp. 2766–2777, 2011. conditions for large-scale systems are derived by catego- [16] F. Castanos, R. Ortega, A. J. van der Schaft, and A. Astolﬁ, “Asymptotic stabilization via control by interconnection of port- rizing agents into symmetry groups and applying local hamiltonian systems,” Automatica, vol. 45, no. 7, pp. 1611–1618, 7 control laws under limited interconnections with neighbors. 2009. Stability for dissipative and passive systems is considered [17] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proceedings of the and conditions are derived for the maximum number of IEEE, vol. 95, no. 1, pp. 215–233, 2007. subsystems that may be added while preserving stability. [18] P. Wu and P. J. Antsaklis, “Symmetry in the design of large-scale There may exist an upper bound on the number of subsys- complex control systems: Some initial results using dissipativity and lyapunov stability,” 18th Mediterranean Conference on Control and tems so to guarantee stability, depending on the structure Automation (MED), pp. 197–202, 2010. of symmetric interconnection. Two kinds of symmetries [19] M. Araki, “Application of m-matrices to the stability problems of are studied in this paper: star-shaped symmetry and cyclic composite dynamical systems,” Journal of Mathematical Analysis and Applications, vol. 52, no. 2, pp. 309–321, 1975. symmetry. These results may be used in the synthesis of [20] S. P. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear large-scale systems with symmetric interconnections. matrix inequalities in system and control theory. SIAM, 1994, vol. 15. [21] P. Tabuada, A. D. Ames, A. Julius, and G. J. Pappas, “Approximate ACKNOWLEDGMENT reduction of dynamic systems,” Systems and Control Letters, vol. 57, We thank Meng Xia for helpful conversations. The no. 7, pp. 538–545, 7 2008. partial support of the National Science Foundation under Grants No. CNS-1035655 and CNS-1446288 is gratefully acknowledged.