Large-Scale Dissipative and Passive Control Systems in Symmetric Shapes

Transcription

1 Large-Scale Dissipative and Passive Control Systems in Symmetric Shapes Technical Report of the ISIS Group at the University of Notre Dame ISIS April 015 V Ghanbari, P Wu and P J Antsaklis Department of Electrical Engineering University of Notre Dame Notre Dame, IN Interdisciplinary Studies in Intelligent Systems Acknowledgements The support of the National Science Foundation under the CPS Grant No CNS is gratefully acknowledged

2 Large-Scale Dissipative and Passive Control Systems in Symmetric Shapes V Ghanbari, P Wu and P J Antsaklis Abstract We study symmetries in the design of large-scale dissipative and passive control In the framework of dissipativity and passivity theory, stability conditions for large-scale systems are derived by categorizing agents into symmetry groups and applying local control laws under limited interconnections with neighbors We show that there exists an upper bound on the number of subsystems that can be added so to preserve stability of dissipative systems Passivity results when delays are present are also derived In cyclic, star-shaped and chain symmetric systems, the subsystems can be heterogeneous as long as they satisfy the same dissipative inequalities Approximate symmetry with respect to interconnections are also considered and the robustness of the results is shown 1 Introduction Symmetry is a basic feature of shapes and graphs and may be found in many real-world networks, such as the Internet and power grid, resulting from the process of tree-like or cyclic growing Since symmetry is related to the concept of a high degree of repetitions or regularities, common in nature, the study of symmetry has been undertaken in many scientific areas, such as in mathematics (Lie groups), in quantum mechanics and in crystallography in Chemistry In the classical theory of dynamical systems, symmetry has also been studied For example, to simplify the analysis and synthesis of large-scale dynamical systems, it is of interest to consider smaller symmetric subsystems with simplified dynamics, which may potentially simplify the analysis in control, planning or estimation[4, 17] When dealing with multi-agent systems with various information constraints and communication protocols, under certain conditions such systems can be expressed as or decomposed into interconnections of lower dimensional systems, which may lead to better understanding of system properties such as stability and controllability[14] Symmetry here means that the system dynamics are invariant under transformations of coordinates Early research on symmetry in dynamical systems may be found in [5, 7, 1] Symmetry in the sense of distributed systems containing multiple instances of identical subsystems are studied in [11, 1], where the controllability of the entire class of systems can be determined by reducing the The authors are with the Department of Electrical Engineering, at the University of Notre Dame, Notre Dame, IN USA ( vghanbar@ndedu; pwu1@ndedu; pantsakl@ndedu) The support of the National Science Foundation under the CPS Grant No CNS is gratefully acknowledged 1

3 model and examining a lower order member of the equivalence class Different forms of symmetry, such as star-shaped or cyclic symmetry[19] give different stability conditions for interconnected systems We will also show this in the present paper using dissipativity theory The notion of dissipativity is a generalization of Lyapunov stability theory where an energy like quantity is defined to be non-increasing over time Dissipative systems can store or dissipate generalized energy supplied to the systems without generating surplus energy The energy is characterized by a storage function 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 function of input, output, state and time Electrical circuits and other physical systems are common examples [18], but the notion of a generalized energy storage function, supply rate and dissipativity can be applied to general nonlinear systems Passivity is a special case of dissipativity, where the supply rate has a special form A passivity property of great interest is that when two passive systems are interconnected in parallel or in feedback configuration, the overall system remains passive Thus passivity may be preserved when large-scale systems are put together from components of passive subsystems Passivity indices may be used to measure the excess or shortage of passivity [3] Dissipativity and passivity in systems, together with decomposition into lower order subsystems have been used before in the study of large-scale systems [14] focuses on Lyapunov stability using vector Lyapunov functions, as well as input-output stability results with dissipative subsystems [13] studies interconnections of dissipative subsystems and specializes to interconnections of special types of dissipative systems, namely finite gain systems, passive systems, and conic systems [8] generalizes previous results to weighted Lyapunov functions and gives spectral characterization of the interconnection matrix Recent papers [1, 15] study the stability conditions in interconnected passive systems, and the close relationships between output feedback passivity and L stability More recent paper [], studied the use of diagonal stability for network of passive systems In this study, stability conditions for large-scale systems consisting of dissipative subsystems are derived by categorizing agents into symmetry groups and applying local control laws under limited interconnections with neighbors Conditions are derived for the maximum number of subsystems that may be added while preserving stability and these results may be used in the synthesis of large-scale systems with symmetric interconnections Early results have appeared in [19] In Section, we introduce needed background on dissipativity, passivity and symmetry in dynamical systems Section 3, 4, and 5 contains stability conditions for symmetric dissipative systems for different symmetry shapes In section 6, symmetry in passive systems with delays has been discussed Section 7 contains simulation results, followed by concluding remarks in Section 8 Mathematical Preliminaries 1 Dissipative and Passive Systems Consider the nonlinear system ẋ(t) = f(x(t)) + g(x(t))u(t) (1) y(t) = h(x(t)) ()

4 where x(t) X R n, y(t) Y R m, u(t) U R m, m n f() R n, g() R n m and h() R m are continuous functions where f(0) = 0, h(0) = 0 and f() is locally Lipschitz Let U be an inner product space whose elements are functions f : R R Also let U m be the space of m-tuples over U, with inner product y, u = m i=1 y i, u i Then for any y(t), u(t) U m and any T R, a truncation u T can be defined via { u(t), for t < T u T (t) = 0, otherwise A truncated inner product is defined as u, v T = u T, v T in an extended space U m e = {u u t U m, T R} The inner product over the interval [0, T ] for continuous time is denoted as y, u T = T 0 yt (t)u(t)dt A system with m inputs and m outputs may now be formally defined as a relation on U m e U m e, that is a set of pairs (u(t) U m e, y(t) U m e ), where u(t) is an input and y(t) the corresponding output Definition 1 [18] A dynamical system with supply rate ω(u, y) is dissipative if there exists a positive definite storage function V (x) such that for all t 1 < t t t 1 ω(u, y)dt V (x(t )) V (x(t 1 )) (3) Definition A system is (Q, S, R) dissipative [13] if the system is dissipative with respect to the supply rate [ ] T [ ] [ ] y Q S y ω(u, y) = u S T (4) R u = y T Qy + y T Su + u T Ru where Q R p p, S R p m, and R R m m, with Q and R symmetric Let Q = I, where I is the unit matrix of appropriate dimension, S = 0 and R = k I, for some fixed positive real number k In this case it can be shown that the above definition (4) implies y T k u T (5) where T is the truncated norm, defined via x T = x, x T If (5) is satisfied, we say that the systems is finite gain input-output stable, or L stable with an upper gain bound of k Consider m dissipative subsystems, i = 1,, m ẋ i = f i (x i ) + g i (x i )u i (6) y i = h i (x i ) (7) each satisfying the dissipativity inequality with supply rate ω i (u i, y i ) = y T i Qy i + y T i Su i + u T i Ru i as in (4) Let the linear feedback interconnections be described by u i = u ei H ij y j, i = 1,, m (8) j=1 3

5 where u i is the input to subsystem i, y i is its output, u ei is an external input, and H ij are constant matrices If we define y = [ y T 1,, yt m] T, H = [Hij ], and define u, u e similarly, then the interconnected system can be represented by u = u e Hy (9) Lemma 1 ([13]) (Small Gain) Let the ith subsystem have finite gain γ i, for i = 1,, m, and suppose that each subsystem has only one input and one output Define Γ = diag {γ 1,, γ N } and A = ΓH with H defined in (8) Then if there exists a diagonal positive definite matrix D such that D A T DA > 0 (10) the interconnected system is BIBO stable Remark 1 : The proof may be found in [13] A sufficient condition for the existence of such P that satisfies (10) is that the matrix  = [â ij ] (11) where â ii = 1 a ii, â ij = a ij, i j with [a ij ] = A defined above ( is an M -matrix [13]) has positive leading principal minors Note that  is positive definite in this case Lemma ([8]) If there exists a diagonal matrix D > 0 such that the matrix ˆQ = H T DRH + DSH + H T S T D DQ (1) is positive definite, ie ˆQ > 0, then the network of m interconnected (Qi, S i, R i )-dissipative agents (6) (7) (8) is asymptotically stable Remark : The proof of this result may be found in [8] This result is a generalization of results in [13] and [14] Lemmas 1 and deal with stability in multi-agent dissipative systems u ei _ u i y i H i1 y 1 H ij y i H im y m Figure 1: Interconnected multi-agent system 4

6 Symmetric Distributed System Consider a dynamical system consisting of subsystems Σ on the diagonal as in Figure 1 Σ is described by ẋ i = f(x i, u i ) y i = h(x i, u i ) (13) with storage function V i (x) and supply rate ω i (u i, y i ), where u i, y i are input/output vectors with the same dimension and i = 1,, m H ij are constant feedback interconnection matrices The system inputs and outputs are stacked as u = [u T 1, ut,, ut m] T, y = [y T 1, yt,, yt m] T ; (ũ, ỹ) is the input and output for the interconnected system We consider three types of symmetries, namely star-shaped symmetry, cyclic symmetry and chain symmetry In a star-shaped symmetric group, subsystems do not have interconnections with each other but the base system has interconnection with all the subsystems In a cyclic symmetric, subsystems have interconnections with their neighbors and the base systems In a chain symmetry, each subsystem has interconnections with its two neighbors, except the leading and ending agents See Figure, 3, 4 Details and results involving such symmetric interconnected systems are given in the next section Subsystem Base System Figure : Star-shaped symmetry Base System Subsystem Figure 3: Cyclic symmetry Leading Agent Ending Agent Figure 4: Chain symmetry 3 Dissipative and Passive Stare-Shaped Symmetric Systems 31 Dissipative System with Star-shaped Symmetric Dissipative Subsystems Consider the nonlinear system 5

7 ẋ = f(x) + g(x)u y = h(x) u = u e Hy (14) where x = [ x T 0, xt 1,, T [ m] xt, y = y T 0, y1 T,, T [ m] yt, u = u T 0, u T 1,, T m] ut Here (x0, y 0, u 0 ) are the states, output and input of the base system; (x i, y i, u i ) with i = 1,, m denote the additional subsystems in (13); H is defined below in (15) The base system is the starting system which we expand upon by adding subsystems H h 1 h 1 h H T 1 H 0 = (15) h T 1 0 H where block entries of H are feedback matrices corresponding to (yj, u i ) as in (8) Specifically, H and h 1 are the interconnection feedback matrices corresponding to (y i, u i ) ie u 0 = u e0 Hy 0 m i=1 h 1y i, u i = u ei h T 1 y i Hy 0, i = 1,, m In the following we focus first on results using the small gain theorem Later in the section we will derive more general results under the assumption of dissipativity and passivity of systems in star-shaped symmetric interconnected systems Theorem 1 (Small Gain) Consider a finite gain system extended by m star-shaped finite gain symmetric subsystems with symmetric interconnection matrix H, if m < σ(â) σ(αâ 1 α T ) (16) where  is the test matrix in (11) and σ and σ are minimum and maximum eigenvalues of the corresponding matrix respectively, then based on Lemma 1, the system with input u and output y is BIBO stable Proof: When the interconnected system is extended with one symmetric subsystem, we have the new test matrix ˆÃ ] = [ˆãij, where ˆã ii = 1 ã ii, ˆã ij = ã ij, i j ã ij is the entries of the matrix [ ] [ ] à = Γ H Γ 0 H h1 = 0 Γ h T 1 H The new test matrix ˆÃ can be written as ˆÃ [ ]  α = α T, where α = Γh 1 since  already has  positive leading principal minors, ˆÃ is an M -matrix if and only if  αâ 1 α T > 0, so that the extended system is still stable Similarly, if the system is extended with m symmetric subsystems, we have the form Γ 0 0 H h 1 h 1 à = Γ H 0 Γ 0 h T 1 H 0 = 0 0 Γ h T 1 0 H 6

8 ˆÃ =  α α α T  0 α T 0  ˆÃ is an M -matrix if and only if  mαâ 1 α T > 0, ie there is an upper-bound of such symmetric extension m < σ(â) σ(αâ 1 α T ) (17) Remark 1 : The term m star-shaped finite gain symmetric subsystems in Theorem 1 refers to m numbers of finite gain subsystems which can be added to the base system with star-shaped symmetry Remark : The existence of such upper bound implies that in order to preserve stability, positive feedback for networked control systems should be restricted Since we assume feedback from symmetric subsystems, their feedback gains should be identical, either all negative feedback or positive feedback For the former case, the system will remain stable even after adding arbitrary number of subsystems The result for negative feedback is a well known result For the latter case, there exist an upper bound on the gain if stability is to be guaranteed, or an upper bound on the number of subsystems that can be added Remark 3 : The proof of this theorem is based on the proof of the small gain theorem in [13], and according to the Lemma 1, all the subsystems are single-input single-output Theorem 1 points to the fact that even with very small gain (Lemma 1), stability may be lost for large numbers of subsystems For the more general case, we consider symmetric subsystems that are (q, s, r) dissipative Consider interconnected subsystems with dynamics shown in (15) Let all subsystems be (q, s, r) dissipative in a star-shaped interconnection as in Figure Note that it is assumed that all subsystems satisfy the dissipative inequalities with the same (q, s, r) Therefore, H b b c h 0 H (18) c 0 h where b and c are feedback gains between the base system and subsystems, ie u 0 = u e0 Hy 0 by i (19) i=1 u i = u ei cy 0 hy i, i = 1,, m (0) 7

9 We now have the following result: Theorem Consider a (Q, S, R) dissipative base system extended by m star-shaped symmetric (q, s, r) dissipative subsystems The whole system is ( ˆQ, Ŝ, ˆR) dissipative with And it is asymptotically stable if ˆQ = S H + H T S T H T R H Q Ŝ = 1 [ H T R + R H] S ˆR = R where σ( m < min( ˆQ) σ(c T rc + β(ˆq b T Rb) 1 β T ), ˆQ = H T RH + SH + H T S T Q > 0 ˆq b T Rb ) (1) ˆq = h T rh + sh + h T s T q > 0 β = Sb + c T s T H T Rb c T rh Proof: For dissipativity: The base system is (Q, S, R) dissipative with supply rate ω(u, y) = y T Qy + y T Su + u T Ru and subsystems are in feedback interconnection with the base system with u = u e Hy where u e is the external input (see figure 1) Then we have V ω(u, y) = y T Qy + y T S[u e Hy] + [u e Hy] T Ru y T Qy + y T Su e y T S Hy + [u T e y T HT ]R[u e Hy] y T Qy + y T Su e y T S Hy + u T e Ru e u T e R Hy y T HT Ru e + y T HT R Hy y T [Q S H + H T R H]y + y T [S H T R R H]u e + u T e Ru e So, ˆQ = S H + H T S T H T R H Q For asymptotic stability: Ŝ = 1 [ H T R + R H] S ˆR = R We have shown that the enlarged system is ( ˆQ, Ŝ, ˆR) dissipative, where ˆQ = diag(q, q,, q),ŝ = diag(s, s,, s), ˆR = diag(r, r,, r) Let D = I, H defined in (18) We require ˆ Q = Ŝ H + H T Ŝ T H T ˆR H ˆQ > 0 () 8

10 This implies ˆ Q = ˆQ mc T rc β β β β T β T Λ β T > 0 (3) where [ A B By Schur s theorem, B T D we have (1) β = Sb + c T s T H T Rb c T rh Λ = ˆq I m b T Rb circ([11 1]) ] > 0 if and only if D > 0 and A BD 1 B T > 0 Thus recursively Remark 1 : The right side of (1) consists of two parts, the first part means the base system will become unstable if the summation of positive feedback from all subsystems exceed certain value, while the second part means the subsystem will become unstable if the positive feedback from the base system is too strong Remark : Notice that in the above we considered all subsystems being (q, s, r) dissipative No restrictions were placed on the actual dynamics which may be different from each other So the above results apply to heterogeneous systems as well, as long as they satisfy the inequality (3) 3 Passive System with Star-shaped Symmetric Passive Subsystems Corollary 1 Consider a passive system extended by m star-shaped symmetric passive subsystems The whole system is asymptotically stable if where ˆQ = H + HT m < σ( ˆQ) σ(β ˆq 1 β T ) > 0, ˆq = h + ht > 0, β = b + ct Proof: To ensure the enlarged system is passive, we require (4) where Q = H + H T > 0 H b b c h 0 H = c 0 h 9

11 This implies symmetry and where Q ˆQ β β β T Λ β T By Schur s theorem, recursively we have (4) [ A B B T D ] β = b + ct > 0 if and only if D > 0 and A BD 1 B T > 0 Thus Remark 1: Note that passive systems are special cases of dissipative systems where Q = 0, S = I, R = 0 for the base system and q = 0, s = I, r = 0 for subsystems One can directly get this result from Theorem (1) Remark : This corollary is a simplified case of the main result in [0], where passivity indices for nonlinear multi-agent systems with feedback interconnections are derived Passivity indices are used to measure the excess and shortage of passivity by rendering the system passive with feedback and feedforward, and describe the performance of passive systems Here instead of deriving passivity indices in explicit from or characterizing passivity indices by a set of matrix inequalities, we are measuring how many subsystems can we add while keeping rendering the system passive Here instead of deriving passivity indices in explicit from or characterizing passivity indices by a set of matrix inequalities, we are measuring how many subsystems can be added while keeping rendering the system passive 33 Approximate Star-shaped Symmetry in Interconnected Passive and Dissipative Systems Exact symmetry in control systems may not apply in some cases, for instance the models of distributed systems may be different but bounded by some induced matrix norm, interconnections may have different weights, there may be time-variant delays, packet drops, etc Disturbance existing in the system dynamics and interconnection structure can represent another type of approximate symmetry The study of approximate symmetry can provide more robust results for dynamical systems There are also a few results of approximate symmetry and approximate model reduction of dynamic systems [16] When introducing approximate symmetry into dissipative systems, the linear feedback inter- 10

12 connections can be described by u 0 = u e0 Hy 0 b i y i i=1 u i = u ei c i y 0 h i y i, i = 1,, m ie star-shaped subsystems have different feedback gains although the structure of the network remains star-shaped symmetric So, we can define the interaction matrix as follows: H = H b 1 b b m c 1 h c 0 h 0 c m 0 0 h m Theorem 3 Consider a dissipative system extended by m star-shaped approximately symmetric dissipative subsystems The whole system is asymptotically stable if { } { } { ˆQ + m Sb1 + c T 1 s T 1 Sb + c T s T Sbm + c T ms T m c i r i c i i=0 H T Rb 1 c T 1 r 1 h 1 H T Rb c T r h H T Rb m c T { } { } mr m h m b T 1 S T + s 1 c 1 s1 h 1 + h T 1 s T 1 b T 1 RH h T 1 r 1 c 1 q 1 b T b T 1 1 Rb Rb b T 1 Rb m { } 1 { } b T S T + s c b T RH h T b T r c Rb s h + h T s T 1 q b T Rb b T { } { m 1 Rb m } b T m S T + s m c m sm b T mrh h T b T mrb 1 b T h m + h T ms T m mrb m 1 mr m c m q m b T mrb m (5) Proof: The proof is similar as in Theorem thus ommited Details are discussed in the below corollary, where passive systems instead of dissipative systems are considered Corollary Consider a passive system extended by m star-shaped approximately symmetric passive subsystems The whole system is asymptotically stable if where ˆQ = H + HT σ( m < ˆQ) maxσ(β i iˆq i 1 βi T i = 1,,, m (6) ), > 0, ˆq i = h i + h T i > 0, β i = b i + c T i 11

13 Proof: We require ˆ Q = S H + H T ST H T R H Q > 0 (7) H + H T b 1 + c T 1 b m + c T m = 1 b T 1 + c 1 h 1 + h T 1 0 > 0 b T m + c m 0 h m + h T m Here due to passivity, Q = R = 0, S = 1 I, thus ˆ Q > 0 if and only if ˆQ i=1 ˆq i = h i + h T i β iˆq 1 i β T i > 0 > 0, i = 1,,, m Note that if ˆq i is positive definite, β iˆq 1 i β T i is also positive definite, thus we have the sufficient condition for asymptotically stability ie, (8) ˆQ m max(β iˆq 1 i β T i ) > 0 (9) σ( m < ˆQ) maxσ(β i iˆq i 1 βi T i = 1,,, m (30) ), 4 Dissipative and Passive Cyclic Symmetric Systems 41 Dissipative System with Cyclic Symmetric Dissipative Subsystems For cyclic symmetry groups, cyclic interconnections are represented by H b b c H h c where h = circ([v 0 v 1 v m 1 ]) is a circulant matrix with first row [v 0 v 1 v m 1 ] Here h = P T hp where P =

14 or h can be written as h = v0 I + v 1 P + v m 1 P m 1 (31) We now consider (14) with (q, s, r) dissipative subsystems, interconnected via a cyclic interconnection as shown in in Figure 3 Theorem 4 Consider a (Q, S, R) dissipative system extended by m cyclic symmetric (q, s, r) dissipative subsystems The whole system is ( ˆQ, Ŝ, ˆR) dissipative with V y T ˆQy + y T Ŝu e + u T e ˆRu e (3) where ˆQ = S H + H T S T H T R H Q > 0 And it is asymptotically stable if Ŝ = 1 [ H T R + R H] S > 0 ˆR = R > 0 where σ( m < min( ˆQ) σ(c T rc + β m Λ 1 βm) T, Λ = rh T h + sh + h T s q I m b T Rb circ([11 1]) ˆq b T Rb ) (33) ˆq = rσ( h)σ( h) + s(σ( h) + σ( h)) q β = Sb + c T s T H T Rb c T r h σ( h) = β m = [ββ β] }{{} m m 1 j=0 m 1 v j λ j i = j=0 v j e πij m Proof: For dissipativity the proof is similar to the proof of Theorem 1 For asymptotic stability: Similarly as with (), we require Q = S H + H T ST H T R H Q = ˆQ mc T rc β β β T Λ β T > 0 (34) where Λ = r[circ(v) T circ(v)] + s[circ(v) + circ(v) T ] (q + b T Rb) I m 13

15 According to Theorem in [8], requiring Λ > 0 is equivalent to assuming (q, s, r)-dissipative agents with interconnection matrix h = circ(v), and ˆQ qst b T Rb circ([11 1]) > 0 Thus the spectral characterization of h should satisfy σ( h) s s < r r q + mbt Rb r (35) m < rσ( h)σ( h) + s(σ( h) + σ( h)) q b T Rb It is known that if two matrices P and Q commute QP = P Q and if λ is a simple eigenvalue of P with eigenvector ν, then ν is also an eigenvector of Q Thus, for every eigenvalue λ i with eigenvector υ i of P, λ i = 1 and λ i = eig(p ) = e πi m, we know that ν i is also a eigenvector of h Multiply both sides of (31) by ν i to obtain Hence hνi = (v 0 I + v 1 P + v m 1 P m 1 )ν i m 1 = (v 0 + v 1 λ i + v m 1 λ m 1 i )ν i = ( v j λ j i )ν i σ( h) = m 1 j=0 j=0 v j λ j i, i = 0, 1,, m 1 The rest of the proof is similar to the proof of Theorem and it is omitted Remark: Similar to (1), the right side of (33) consists of two parts, indicating the small-gain like property that the positive feedback between the base system and subsystems should not be too strong However, unlike the star-shaped structure, the ring-shaped structure does not necessarily need the support of the base system, which means no constraints are put on m as in (33) if there is no base system In this case, H h = circ(v), and (35) is reduced to where σ( h) = σ( h) s s < r r q r m 1 j=0 m 1 v j λ j i j=0 v j λ j i m 1 j=0 v j (36) σ( h) is bounded and the bound will not be affected by the number of subsystems given a similar structure of the interaction matrix, in which zero entries are filled when adding new symmetric subsystems 14

16 4 Passive System with Cyclic Symmetric Passive Subsystems Corollary 3 Consider a passive system extended by m cyclic symmetric passive subsystems The whole system is asymptotically stable if m < σ( ˆQ) σ(β m Λ 1 β T m) (37) where Λ = h + ht σ( h) = β = b + ct β m = [ββ β] }{{} m m 1 j=0 m 1 v j λ j i = j=0 v j e πij m Proof: Same as (), we require Q = S H + H T ST H T R H Q = ˆQ mc T rc β β β T Λ β T > 0 (38) According to Proposition in [9], requiring Λ > 0 is equivalent to assuming (q, s, r) dissipative agents with interconnection matrix h = circ(υ), and ˆQ qsr b T rb circ([111]) > 0 thus the spectral characterization of h should satisfy σ( h) > 0 It is known that if two matrices P and Q commute, so that QP = P Q and if λ is a simple eigenvalue of P with eigenvector ν, then ν is also an eigenvector of Q Thus, for every eigenvalue λ i with eigenvector ν i of P, λ i = 1 and λ i = eig(p) = e πi m, we know that ν i is also an eigenvector of h Multiply by ν i at the both side of h to obtain symmetry and hν i = (υ 0 I + υ 1 P + υ m 1 P m 1 )ν i m 1 = (υ 0 + υ 1 λ i + + υ m 1 λ m 1 i )ν i = ( υ j λ j i )ν i j=0 Hence σ( h) = m 1 j=0 υ j λ j i, i = 1,, m 1 15

17 The rest of the proof is similar as in Theorem 1 thus omitted Remark: Comparing to Theorem 3, the condition (36) on the spectral characterization of h is not required here The reason is that passive systems are a special case of (q, s, r) dissipative systems with q = 0, s = I, r = 0, therefore the spectral characterization can always satisfy the stability condition 5 Dissipative and Passive Chain Symmetric Systems 51 Dissipative System with Chain Symmetric Dissipative Subsystems Now we are considering (14) with (q, s, r) dissipative subsystems, and a chain interconnection as shown in in Figure 4 Here H b b b c h e H = c f h e c f h Theorem 5 Consider a (Q, S, R) dissipative system extended by m chain-connected symmetric (q, s, r) dissipative subsystems The whole system is asymptotically stable if Q = S H + H T ST H T R H Q > 0 where Q = diag(q, q,, q), S = diag(s, s,, s), R = diag(r, r,, r), ˆQ mc T rc Sb + c T s H T Rb c T rh c T rf Sb + c T s H T Rb c T re c T rh c T rf sc + b T S T b T RH h T rc f T rc sh + h T s T q b T Rb h T rh f T rf se + f T s T b T Rb h T re f T rh sc + b T S T b T RH e T rc h T rc f T rc sf + e T s T b T Rb e T rh h T rf sh + h T s T q b T Rb e T re h T rh f T rf b T Rb e T rf sc + b T S T b T RH e T rc h T rc f T rc sc + b T S T b T RH e T rc h T rc b T Rb Sb + c T s H T Rb c T re c T rh c T rf Sb + c T s H T Rb c T re c T rh b T Rb f T re b T Rb b T Rb f T re sh + h T s T q b T Rb e T re h T rh f T rf se + f T s T b T Rb h T re f T rh b T Rb e T rf sf + e T s T b T Rb e T rh h T rf sh + h T s T q b T Rb e T re h T rh Proof: The proof is similar to the proof of Theorem and 4 thus omitted > 0 (39) Remark:The chain-connected symmetric subsystems are shown in Figure 4 Since the structure in the LMI (39) is not as straightforward as in (3) and (38), the restriction on m is not derived explicitly 16

18 5 Passive System with Chain Symmetric Passive Subsystems Corollary 4 Consider a passive system extended by m chain-connected symmetric SISO passive subsystems The whole system is asymptotically stable if m < σ( H+HT ) ββ T (h e f) where or Q = H b b b c h e H = c f h e c f h H + H T b + c T b + c T b + c T c + b T h + h T e + f T 0 c + b T f + e T h + h T e + f T c + b T 0 f + e T h + h T > 0 (40) Remark: The LMI (40) is the simplified form of (39) 6 Symmetry in Passive Systems with Delays In this section, we focus on stability results of passive systems with delays Normally dissipativity and passivity cannot be preserved if random delays are introduced into the network To deal with constant delays, the wave variable transformation defined in [10, 6] is utilized Consider star-shaped symmetry interconnections with constant link delays T i,j between agent i and agent j The interconnection matrix corresponding to Figure 6 is H h 0 H 1 0 h The constant delays can be modeled as u i (t) = u i1 (t T 0i ) v i (t) = v i1 (t T i0 ) The wave variable transformation for each input and output pair is given by [ ] [ ] [ ] ui1 I bi = vi1 bi bi y id 17 y 0

19 Figure 5: Multiple Passive Systems with Delays 18

20 [ vi y 0d ] = I 1 b I b I b I [ ui y i where b is a constant The energy V Ni stored in the delayed network between agent 0 and agent i is non-negative, because V Ni t t 0 (u T i1u i1 + v T iv i u T iu i v T i1v i1 )dτ ] Therefore t = t t u T i1u i1 dτ + viv T i dτ 0 t T 0i t T i0 t 0 (u T i1u i1 v T i1v i1 )dτ = T t 0 y T 0 y id dτ t t 0 (v T iv i u T iu i )dτ T t 0 y T i y 0d dτ Assume the base system Σ 0 is output feedback passive(ofp) with passivity index ρ 0 > 0, and passive subsystems Σ i with OFP index ρ i = ρ Then according to definition V (x i ) y T i u i ρ i y T i y i where u 0 = u e0 m i=1 y id Hy 0, u i = u ei + y 0d hy i Define the storage function V (x) = V (x i ) for input-output pair u e = [ u T e0,, ut em] T and y = [ y T 0,, yt m] T Then = yi T u ei i=0 V (x) yi T u i ρ i yi T y i i=0 ρ i yi T y i y0 T Hy 0 i=0 yi T u ei i=0 i=0 yi T hy i i=1 ρ i y T i y i λ( H + HT yi T u ei i=0 )y T 0 y 0 γyi T y i i=0 y0 T y id + i=1 i=1 λ( h + ht yi T y id i=1 )y T i y i where γ = min(ρ 0 + λ( H+HT ), ρ + λ( h+ht )) Hence the interconnected system with constant network delay has the OFP index γ Remark: If there is no delay in the network and (4) is applied, it can be shown that the upper bound for m is infinity Restrictions are put on the spectrum of H and h, which confirms the results in this section 19

21 7 Examples The following examples illustrate the above results We show that stability conditions for large-scale systems can be derived by categorizing agents into symmetry groups and applying local control laws under limited interconnections with neighbors There may exist an upper bound on the number of subsystems so to guarantee stability, depending on the structure of symmetric interconnection Example 1: Suppose we have m + 1 finite gain symmetric subsystems in a star symmetry as in Figure 3, each of gain less or equal to 1, and an interconnection matrix H = } {{ 01 } m+1 The problem is to find how large can m be, ie how many subsystems can be connected in symmetric interconnections without losing stability For such system, we know Q = I, S = 0, R = 1 4I According to Theorem (1), Thus m max = 3 m < min(311, 65) = 311 In fact, when m = 3, the interconnected system is finite gain input-output stable, since ˆ Q = > But when m = 4, ˆ Q = It is not positive definite, and stability can no longer be guaranteed Example : We are considering a similar example as in [8], the rendezvous of multiple agents with damping and inertia The dynamics for each agent are M z + Bż = u, y = z 0

22 where z, ż R is the position and velocity, M and B are positive definite matrices, represent the uncertain positive definite inertia and damping matrices, respectively Consider the storage function V = ż T M T B 1 Mż + ż T Mz + 1 zt Bz Each agent is (q, s, r) dissipative with q = 0, s = 1 I, r = B 1 MB 1 > 0 Assume there is an agent in the center while the other agents have cyclic interconnection structure, see Figure 7 Figure 6: Cyclic interconnection structure with a center According to Theorem 3 (35), the system is stable if σ( h) s s < r r q + mbt Rb r < s r But not all σ( h) satisfy the condition above, thus the stability condition is not satisfied In fact simulation results show that such system is always unstable, no matter how many agents are added and how small b T Rb is, see Figure 8 Example 3: In Figure 9 we remove the center and consider a network of cyclic symmetric subsystems Each subsystem is (q, s, r) dissipative with q = 1, s = 0, r = 4, with interaction matrix H = h = }{{} 1

23 Figure 7: Trajectories of extended agents Figure 8: Cyclic interconnection structure without a center

24 According to Theorem 3 (35), the system is stable if σ( h) s m 1 = r v j e πij m s 03 < 05 = r q r j=0 The inequality above always holds Thus the system can be extended with infinite numbers of subsystems without losing stability Example 4 (approximate symmetry): Suppose we have m + 1 symmetric passive subsystems in Figure 3, and an interconnection matrix H = } {{ 1 } m+1 In order to preserve stability for the interconnected matrix, we apply Corollary 4 (30) to determine how many subsystems can be connected in symmetric interconnections without losing stability σ( m < ˆQ) maxσ(β i iˆq i 1 βi T ) = 65 Thus m max = 6, which is relatively conservative as an upper bound of the number of subsystems 8 Conclusion In this paper, stability conditions for large-scale systems are derived by categorizing agents into symmetry groups and applying local control laws under limited interconnections with neighbors Stability for dissipative and passive systems is considered and conditions are derived for the maximum number of subsystems that may be added while preserving stability These results may be used in the synthesis of large-scale systems with symmetric interconnections References [1] M Arcak and E D Sontag A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks 46th IEEE Conference on Decision and Control, pages , 007 [] Murat Arcak Diagonal stability on cactus graphs and application to network stability analysis Automatic Control, IEEE Transactions on, 56(1): , 011 3

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