Passivity Indices for Symmetrically Interconnected Distributed Systems

Transcription

1 9th Mediterranean Conference on Control and Automation Aquis Corfu Holiday Palace, Corfu, Greece June 0-3, 0 TuAT Passivity Indices for Symmetrically Interconnected Distributed Systems Po Wu and Panos J Antsaklis Abstract In this paper, the passivity indices for both linear and nonlinear multi-agent systems with feedforward and feedback interconnections are derived For linear systems, the passivity indices are explicitly characterized, while the passivity indices in the nonlinear case are characterized by a set of matrix inequalities We also focus on symmetric interconnections and specialize the passivity indices results to this case An illustrative example is also given I INTRODUCTION Passive systems can be thought of as systems that do not generate energy, but only store or release the energy which was provided The notion of energy here is a generalization from the traditional notion of energy in physics and is characterized by a storage function Passivity is a special case of dissipativity, which can be applied to linear and nonlinear systems The benefit of passivity is that when two passive systems are interconnected in parallel or in feedback, the overall system is still passive Thus passivity is preserved when large-scales systems are combined from components of passive subsystems 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, The concepts of passivity and dissipativity are introduced in, 3, 4 There have been early studies on interconnected systems5 Recent papers 6, 7 study the stability conditions in interconnected passive systems, and the close relationships between output feedback passivity and L stability 8 designs a passivity-based controller for asymptotic stabilization of interconnected port-hamiltonian systems Passivity indices of cascade systems are measured in 9 Symmetry, as one basic feature of shapes and graphs, has been exhibited 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, the study of symmetry has been appealing in many scientific areas, such as Lie groups in quantum mechanics and crystallography in chemistry In the classical theory of dynamical systems, symmetry has also been extensively studied For example, to simplify The authors are with the Department of Electrical Engineering, at the University of Notre Dame, Notre Dame, IN USA pwu@ndedu; pantsakl@ndedu) the analysis and synthesis of large-scale dynamic systems, it is always of interest to reduce the dynamics of a system into smaller symmetric subsystems, which potentially simplifies control, planning or estimation tasks When dealing with multi-agent systems with various information constraints and 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 Then the existence of symmetry here means that the system dynamics are invariant under transformations of coordinates Early research on symmetry in dynamical systems could be found in 0,, Symmetry in the sense of distributed systems containing multiple instances of identical subsystems are studied in 3, 4, where the controllability of the entire class of systems can be determined by reducing the model and examining a lower order member of the equivalence class Different forms of symmetry, such as star-shaped or cyclic symmetry5 give different stability conditions for interconnected systems We will also show that in this paper The current paper is motivated by the interest of sufficient stability conditions in 5 and passivity as a binary system characterization Passivity indices used in can measure the level of passivity thus imply the degree of stability With the distributed setup in this paper, the passivity indices for both linear and nonlinear multi-agent systems with feedforward and feedback interconnections are derived For linear systems, the passivity indices are explicitly characterized, while the passivity indices in the nonlinear case are characterized by a set of matrix inequalities We also focus on symmetric interconnections and specialize the passivity indices results to this case An illustrative example is also given The paper is organized as follows In Section II, we introduce some background on passivity indices and symmetry in dynamical systems In Section III, the passivity indices for interconnected systems are derived for both linear and nonlinear systems Section IV deals with symmetric interconnections Section V contains simulation results, followed by concluding remarks in Section VI II PRELIMINARIES AND BACKGROUND A Passive and Dissipative Systems Consider the nonlinear system ẋ fx, u), y hx, u) ) //$600 0 IEEE

2 where x X R n, y Y R m, u U R m, m n Let U be an inner product space whose elements are functions u : R R Also let U m be the space of n- tuples over U, with inner product y, u m i y i, u i Then for any y, u U m and any T R, a truncation u T can be defined via u T t) { ut), for 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)ut)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 U m e, y U m e ), where u is an input and y the corresponding output Definition A system is dissipative if there exists a positive definite storage function V x) such that for some supply rate ωu, y) and for all t < t t t ωu, y)dt V x t)) V x t)) 3 A system is Q, S, R) dissipative if the system is dissipative with respect to the supply rate T y Q S y ωu, y) u S T R u Consider the nonlinear system and the local group of transformations on X U defined by X, U) φ g x), ψ g u)) where g G, φ g acting on X and ψ g acting on U are local diffeomorphisms Definition 6The system has a G-symmetry or is G- invariant if fφ g x), ψ g u)) Dφ g x) fx, u) for all g, x, u Figure Symmetry in geometric control Definition 3 6The output y hx, u) is G-equivalent if there exists a transformation groupρ g ) g G on Y such that fφ g x), ψ g u)) ρ g hx, u) for all g, x, u With X, U) φ g x), ψ g u)) and Y ρ g y), the properties can be rewritten as Ẋ fx, U), Y hx, U) It means the system remains unchanged under the transformation The two previous definitions can be illustrated by the commutative diagram in Fig C Symmetric Distributed System y T Qy + y T Su + u T Ru where Q R p p, S R p m, and R R m m be constant matrices, with Q and R symmetric A system is passive and simultaneous input feedforward passiveifp) and output feedback passiveofp) if the system is dissipative with respect to the supply rate ωu, y) + ρν)y T u νu T u ρy T y where ρ R is the OFP index and ν R is the IFP index When ρ ν 0, ωu, y) y T u The system is said to be passive When ρ 0, ν 0, ωu, y) y T u νu T u The system is said to be input feedforward passiveifp) 3 When ρ 0, ν 0, ωu, y) y T u ρy T y The system is said to be output feedback passiveofp) In the case when ν > 0 or ρ > 0, the system is said to be input strictly passiveisp) or output strictly passiveosp) respectively Obviously passivity is a special case of dissipativity Passivity indices ρ and ν can be generalized to be ρy) and νu) if the feedforward and the feedback is not static B Symmetry in Dynamical Systems Intuitively, symmetry in dynamical systems means that the system dynamics are invariant under transformations We formalize this in terms of the concepts of diffeomorphism Figure Interconnected multi-agent system Consider a multi-agent dynamical system consisting of subsystems Σ on the diagonal as in Fig Σ is described by ẋ i fx i, u i ) ) y i hx i, u i ) with storage function V i x), supply rate ω i u i, y i ), where i,, m H ρ and H ν are constant feedback and feedforward interconnection matrices, the system inputs and outputs are stacked u u T, u T,, u T m T, y y T, y T,, y T m T ũ, ỹ) is the input and output for the interconnected system To describe the relationships between multiple agents, we //$600 0 IEEE

3 have a digraph G to model the interaction topology for a network control system 7 Here, for a symmetric group S p consisted of p subsystems, we consider three types of symmetries, namely star-shaped symmetry, cyclic symmetry and chain symmetry Intuitively, in a star-shaped symmetric group, subsystems do not have interconnections with each other In cyclic symmetric group, subsystems contribute to a close related automorphism group In chain symmetry, each subsystem has interconnections with its two neighbors, except the leading and ending agents See Fig 3, 4, 5 Fig3 Star-shaped symmetry Fig5 Chain symmetry Fig4 Cyclic symmetry III PASSIVITY INDICES FOR INTERCONNECTED SYSTEMS Let subsystem Σ be a passive system with OFP index ρ and IFP index ν It can be easily shown that the stacked system is also passive with indices ρ, ν) If we define the storage function as V x) m i V ix), then V x) is bounded because V x) V i x) ω i u i, y i ) i i i i + ρν) yi T u i ν u T i u i ρ yi T y i i + ρν)y T u νu T u ρy T y ωu, y) So ωu, y) is the supply rate for the multi-agent system With feedback interconnection matrix H ρ and feedforward interconnection matrix H ν see Fig ), the new input and output is given by { ỹ y Hν u 3) ũ u H ρ y Next we are deriving the passivity indices of the interconnected distributed system with respect to input-output pair ũ, ỹ) A Linear Passive Systems Theorem For a minimum phase linear system Gs) consisting of scalar passive subsystems, with only the output feedback loop, ie H ρ 0, H ν 0, the output feedback passivity index is given by T ρ ρ λ 4) Proof: According to, the OFP index is defined as ρ ρ Gs)) min λ G jω) + G jω) ) 5) where λ ) is the minimum eigenvalue of a matrix For the closed loop system which is also minimum phase G cl s) I GH ρ ) G G H ρ ) the new OFP index is given by ρ ρ G cl s)) min λ G cl jω) + ) G cl jω) min λ G jω) + G jω) Hρ Hρ It is known 8 that if two matrices A and B commute, so that AB BA, A, B R n, then the two sets of eigenvalues {λ i A+B)} and {λ i A)+λ i B)} are equal; also λa+b) λa) + λb) Since all scalar subsystems have the same dynamic, G jω) + G jω) is a diagonal matrix with identical diagonal entries, which commutes with any matrix Thus ρ min λ G jω) + G jω) ) + λ H ρ Hρ ) T ρ λ When Gs) is not diagonal, we can derive the boundary of ρ From Fact 58, if A and B are Hermitian matrices, λa) + λ B) λa B) λa) λb) Since here G jω) + G jω) and Hρ Hρ are both Hermitian matrices, ρ min λ G jω) + G jω) ) λ H ρ + Hρ ) ) T ρ λ ρ min λ G jω) + G jω) ) +λ H ρ Hρ )) ρ λ T ie for any G and H ρ, ρ λ T T ρ ρ λ ) //$600 0 IEEE 3

4 Theorem For a linear stable system Gs) consisting of scalar passive subsystems, with only the input feedforward loop, ie H ρ 0, H ν 0 The input feedforward passivity index is Hν + H ν ν ν λ 6) Proof: According to, for a linear stable system Gs), the IFP index can be calculated as ν ν Gs)) min λ Gjω) + G jω)) for the closed loop system which is also stable the new IFP index is G cl s) G H ν ν min λ G cljω) + G cljω)) min λ Gjω) + G jω) H ν Hν ) Hν + H ν ν λ A similar approach can be found in 9 as a passivation method Also, if G is not diagonal, the following inequality holds ν λ Hν + Hν Hν + H ν ν ν λ B Nonlinear Systems Theorem 3 Consider a nonlinear multi-agent system with the input feedforward interconnection matrix H ν and output feedback interconnection matrix H ρ, where I H ν H ρ and I H ρ H ν are nonsingular If the subsystems have the same passivity indices ρ, ν), then the passivity indices ρ, ν) for the whole system satisfy where H P P P H T P H 0 7) +ρν ρi I + S I S T νi + ρ ν ρi I + S I S T νi +ρν + ρ ν I H ν H ρ ) I H ν H ρ ) H ν I H ρ H ν ) H ρ I H ρ H ν ) S and S can be any matrix Proof: Write 3) in matrix form ỹ y H ũ u where H I H ρ H ν I Let H H, then y u H ỹ ũ where H can be calculated as H H I H ν H ρ ) I H ν H ρ ) H ν I H ρ H ν ) H ρ I H ρ H ν ) From the definition of passivity index T V x) + ρν)u T y νu T u ρy T y y y P u u where S can be any matrix Then T y y ỹ V x) P u u ũ T HT P H ỹ ũ For the new passivity index pair ρ, ν), we are requiring a more loose boundary for V x) T V x) ũ T ỹ νũ T ũ ρỹ T ỹ ỹ ỹ P ũ ũ where ρ and ν are the largest value to make the inequality holds Thus we are requiring P H T P H 0 Note that in the nonlinear case, passive subsystems are no longer required to be scalar systems Normally there is no analytic way to derive ρ and ν separately from the matrix inequality 7), but there are special cases: ) OFP case Only consider a feedback loop and ρ In this case, H ν 0, I 0 I 0 ν ν 0, H, H H H ρ I H ρ I For any S, let S S, then P H T P H ρi I + S I ST 0 I H T ρ ρi I + S I 0 0 I I ST 0 H ρ I ρi ρi H ρ + Hρ SH ρ + Hρ T S T 0 Since SH ρ + H T ρ S T is a skew symmetric matrix, x T SH ρ + H T ρ S x 0 for all x R n, which means we are requiring or ρi ρi H ρ + H T ρ ) 0 ρ ρ λ Hρ + H T ρ ) //$600 0 IEEE 4

5 Because passivity index is defined as the maximal possible value, the new OFP index ρ of the interconnected distributed system with respect to input-output pair ũ, ỹ) is T ρ ρ max ρ λ 8) which is the same as 4) If we let ρ > 0, E H ρ, S 0, then ρ can be positive if E is diagonally stable with D I, that is, DE + E T D < 0 Since output passivity implies L stability, this result shows that a family of L stable systems with a feedback connection can also be L stable, given E diagonally stable This conclusion can also be found in 6 and 7, where each subsystem has a distinct passivity index γ i instead of a uniformed passivity index ρ as here ) IFP case Only consider feedforward loop and ν In this case, H ρ 0, ρ ρ 0, H I Hν 0 I I Hν 0 I For any S, let S S, then, H H P H T P H 0 I + S I ST νi I 0 0 Hν T I + S I Hν I I ST νi 0 I 0 νi νi H ν + Hν Hν T S + S T H ν where H T ν S S T H ν is a skew symmetric matrix Similarly, we are requiring ie νi νi H ν + H T ν ) 0 ν ν λ Hν + Hν thus the new IFP indices of the interconnected distributed system with respect to input-output pair ũ, ỹ) is ν ν max ν λ Hν + Hν 9) which is the same as 6) IV SYMMETRIC INTERCONNECTIONS Symmetry existing in the structure of interconnection may reduce the complexity of system analysis For instance, if the symmetry is linear symmetry, ie a diffeomorphism φ g x) is on a sub-group of general linear group GLn, R n ), then the properties of matrices can be applied instantly A permutation of a set X {,, p} is a one-to-one mapping of X onto itself Such a permutation is written, p k k k p which represents that is mapped to k, is mapped to k, etc Such permutation can be represented by φ g x) P x, where P is a orthogonal matrix with only one entry equal to in each row and column, and other entries 0 If all subsystems have the same dynamic and the interconnection matrix is invariant under the permutation, then a symmetry exists in the multi-agent system Since the measurement of passivity indices only relies on the eigenvalues of interconnection matrices, we can examine the matrix properties and give a more explicit result A Cyclic Symmetry Cyclic interconnection matrix H circv 0 v v m ) is invariant under cyclic permutation We can calculate the eigenvalue of H+HT to derive explicit values for the passivity indices where λh) m j0 v j λ j i, i 0,,, m λ i e πi m The sum of two cyclic matrices is still a cyclic matrix H + H T H + H T λ circv 0 v + v m m v 0 + j0 v j + v m j vm + v ) Then we can derive 8) and 9) more explicitly B Star-shaped Symmetry e πij m, i 0,, m Intuitively, in a star-shaped symmetric group, subsystems do not have interconnections with each other, but commute with the center agent with different connection weights h b b m c h 0 H c m 0 h Then H + H T H + H T λ b +c b m +c m h b +c h 0 b m +c m 0 h h ± m b i + c i ), ) i which will be shown as an example in Section V //$600 0 IEEE 5

6 C Chain Symmetry In chain symmetry, each subsystem has interconnections with its two neighbors, except the leading and ending agents b c 0 H T ra, b, c) a b c 0 a b a λh) b + c c cos λ H + HT H + H T iπ m +, i,, m T r a + c, b, a + c ) ) b + a + c) cos V EXAMPLE iπ m +, i,, m Given a minimun phase linear multi-agent system G with only output feedback interconnection matrix H ρ, s+ 0 s+ Gs) 0 s+ 0 s H ρ From 5) and 0) λ H ρ + H T ρ ) 4747 From Nyquist plots of G jω) and G cl jω), ρ min λ G jω) + G jω) ) ρ min λ G cl jω) + ) G cl jω) 3747 thus it can be verified that ρ ρ λ H ρ + Hρ Here ρ 3747 < 0 means the multi-agent system G is short of passivity after the feedback loop is closed But if H ρ then passivity is preserved because ρ ρ λ H ρ + H T ρ ) > 0 VI CONCLUSIONS In this paper, the passivity indices for both linear and nonlinear multi-agent systems with feedforward and feedback interconnections are derived For linear systems, the passivity indices are explicitly characterized, while the passivity indices in the nonlinear case are characterized by a set of matrix inequalities We also focus on symmetric interconnections and specialize the passivity indices results to this case An illustrative example is also given REFERENCES J Bao and P Lee, Process control : the passive systems approach London: Springer-Verlag, 007 J C Willems, Dissipative dynamical systems part i: General theory, Archive for Rational Mechanics and Analysis, vol 45, no 5, pp 3 35, 97 3 P Moylan and D Hill, Stability criteria for large-scale systems, Automatic Control, IEEE Transactions on, vol 3, no, pp 43 49, M J McCourt and P J Antsaklis, Connection between the passivity index and conic systems, ISIS Technical Report, M Sundareshan and M Vidyasagar, L-stability of large-scale dynamical systems: Criteria via positive operator theory, Automatic Control, IEEE Transactions on, vol, no 3, pp , M Arcak and E D Sontag, A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks, Decision and Control, th IEEE Conference on, pp , E D Sontag and M Arcak, Passivity-based stability of interconnection structures, vol 37, pp 95 04, F Castanos, R Ortega, A J van der Schaft, and A Astolfi, Asymptotic stabilization via control by interconnection of port-hamiltonian systems, Automatica, vol 45, no 7, pp 6 68, H Yu and P J Antsaklis, A passivity measure of systems in cascade based on passivity indices, Decision and Control, 00 49th IEEE Conference on, pp 86 9, 00 0 J Grizzle and S Marcus, The structure of nonlinear control systems possessing symmetries, Automatic Control, IEEE Transactions on, vol 30, no 3, pp 48 58, 985 S Y Zhang, The structures of symmetry and similarity of complex control systems, Control Theory and Applications, vol, no, pp 3 37, 994 F Fagnani and J C Willems, Representations of symmetric linear dynamical systems, SIAM Journal on Control and Optimization, vol 3, no 5, pp 67 93, M B McMickell and B Goodwine, Motion planning for nonlinear symmetric distributed robotic formations, The International journal of robotics research, vol 6, no 0, pp 05 04, 007 4, Reduction and nonlinear controllability of symmetric distributed robotic systems with drift, Robotics and Automation, 00 Proceedings ICRA 0 IEEE International Conference on, vol 4, pp , 00 5 P Wu and P J Antsaklis, Symmetry in the design of large-scale complex control systems: Some initial results using dissipativity and lyapunov stability, Control and Automation MED), 00 8th Mediterranean Conference on, pp 97 0, 00 6 S Bonnabel, P Martin, and P Rouchon, Symmetry-preserving observers, Automatic Control, IEEE Transactions on, vol 53, no, pp 54 56, P J Antsaklis and J Baillieul, Special issue on technology of networked control systems, Proceedings of the IEEE, vol 95, no, pp 5 8, D S Bernstein, Matrix mathematics: theory, facts, and formulas Princeton, New Jersey: Princeton University Press, W Z Zhang, J Bao, and P L Lee, Decentralized unconditional stability conditions based on the passivity theorem for multi-loop control systems, Industrial Engineering Chemistry Research, vol 4, no 6, pp , //$600 0 IEEE 6