Abstract
The system-theoretic study of interconnected systems is not new. It started with the work by Gilbert (1963) on controllability and observability for generic classes of systems in parallel, series, and feedback interconnections. Complete characterizations for multivariable linear systems were obtained by Callier and Nahum (1975) for series and feedback interconnections and in a short note by Fuhrmann (1975) for parallel interconnections. We refer the reader to Chapter 10 for a proof of these classical characterizations using the techniques developed here. However, the interconnection structures of most complex systems are generally not of the series, parallel, or feedback type. Thus, one needs to pass from the standard interconnections to more complex ones, where the interconnection pattern between the node systems is described by a weighted directed graph. This will be done in the first part of this chapter. The main tool used is the classical concept of strict system equivalence. This concept was first introduced by Rosenbrock in the 1970s for the analysis of higher-order linear systems and was subsequently developed into a systematic tool for realization theory through the work of Fuhrmann. Rosenbrock and Pugh (1974) provided an extension of this notion toward a permanence principle for networks of linear systems. Section 9.2 contains a proof of a generalization of this permanence principle for dynamic interconnections. From this principle we then derive our main results on the reachability and observability of interconnected systems. This leads to very concise and explicit characterizations of reachability and observability for homogeneous networks consisting of identical SISO systems. Further characterizations of reachability are obtained for special interconnection structures, such as paths, cycles, and circulant structures.
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References
B. Bamieh, F. Paganini, M. Dahleh, “Distributed control of spatially invariant systems”, IEEE Trans. Autom. Control, vol. 47 (7), (2002), 1091–1107.
R.R. Bitmead and B.D.O. Anderson, “The matrix Cauchy index: Properties and applications”, SIAM J. Control Optim., vol. 33(4), (1977), 655–672.
S. Bittanti and P. Colaneri, Periodic Systems: Filtering and Control, Communications and Control Engineering Series, Springer-Verlag, London, 2010.
R. Bott and R.J. Duffin, “Impedance synthesis without the use of transformers”, J. Applied Physics, vol. 20, (1940), 816.
R.W. Brockett, “Linear feedback systems and the groups of Galois and Lie”, Linear Algebra Appl., vol. 50, (1983), 45–60.
R.W. Brockett, “On the control of a flock by a leader”, Proc. Steklov Institute of Mathematics, vol. 268 (1), (2010), 49–57.
R.W. Brockett and J.L. Willems, “Discretized partial differential equations: examples of control systems defined on modules”, Automatica, vol. 10, (1974), 507–515.
O. Brune, “Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency,” J. Math. Phys., vol. 10, (1931), 191–236.
C.I. Byrnes and T. Duncan, “On certain topological invariants arising in system theory”, in New Directions in Applied Mathematics, P. Hilton and G. Young, Eds., Springer, Heidelberg, 29–71, 1981.
F.M. Callier, and C.D. Nahum, “Necessary and sufficient conditions for the complete controllability and observability of systems in series using the coprime decomposition of a rational matrix”, IEEE Trans. Circuits Syst., vol. 22, (1975), 90–95.
J.-M. Dion, C. Commault and J. van der Woude, “Generic properties and control of linear structured systems: A survey”, Automatica, vol. 39, (2003), 1125–1144.
P.A. Fuhrmann, “On controllability and observability of systems connected in parallel”, IEEE Trans. Circuits Syst., vol. 22 (1975), 57.
P.A. Fuhrmann and U. Helmke, “Strict equivalence, controllability and observability of networks of linear systems”, Math. Contr. Signals and Systems, vol. 25, (2013), 437–471.
E.G. Gilbert, “Controllability and observability in multivariable control systems”, SIAM J. Control Optim., vol. 2, (1963), 128–151.
S. Hara, T. Hayakawa and H. Sugata, “LTI systems with generalized frequency variables: A unified framework for homogeneous multi-agent dynamical systems”, SICE J. of Contr. Meas. and System Integration, vol. 2, (2009), 299–306.
M. Hazewinkel and C.F. Martin, “Symmetric linear systems: an application of algebraic systems theory”, Int. J. of Control, vol. 37 (1983), 1371–1384.
U. Helmke, “Rational functions and Bezout forms: a functorial correspondence”, Linear Algebra Appl., vol. 122–124, (1989), 623–640.
T.H. Hughes and M.C. Smith, “Algebraic criteria for circuit realizations”, in Mathematical Systems Theory, A Volume in Honor of U. Helmke, (K. Hüper and J. Trumpf; Eds.), CreateSpace, 211–228, 2013.
J.Z. Jiang, and M.C. Smith, “Regular positive-real functions and five-element network synthesis for electrical and mechanical networks”, IEEE Trans. Autom. Control, vol. 56(6), (2011), 1275–1290.
R.E. Kalman, “Old and new directions of research in system theory”, Perspectives in Mathematical System Theory, Control and Signal Processing, Eds. J.C. Willems, S. Hara, Y. Ohta and H. Fujioka, Springer, vol. 398, 3–13, 2010.
S. Koshita, M. Abe, and M. Kawamata, “Analysis of second-order modes of linear discrete-time systems under bounded-real transformations”, IEICE Trans. Fundamentals, vol. 90, (2007), 2510–2515.
E.L. Ladenheim, “A synthesis of biquadratic impedances”, M.S. thesis, Polytechnic Institute of Brooklyn, 1948.
C.-T. Lin, “Structural controllability”, IEEE Trans. Autom. Control, vol. 19 (3), (1974), 201–208.
Y.Y. Liu, J.-J. Slotine, A.L. Barabasi, “Controllability of complex networks”, Nature, vol. 473 (2011), 167–173.
J. Lunze, “Dynamics of strongly coupled symmetric composite systems”, Int. J. Control, vol. 44, (1986), 1617–1640.
P. Müller, “Primitive monodromy groups of polynomials”, Recent developments in the inverse Galois problem, M. Fried (Ed.), Contemp. Maths. vol. 186, Am. Math. Soc., 385–401, 1995.
C.T. Mullis and R.A. Roberts, “Roundoff noise in digital filters: frequency transformations and invariants”, IEEE Trans. Acoust. Speech Signal Processing, vol. ASSP-24, (1976), 538–550.
F. Pakovich, “Algebraic curves P(x) − Q(y) = 0 and functional equations”, Complex Var. Elliptic Equ., vol. 56, (2011), 199–213.
G. Parlangeli and G. Notarstefano, “On the reachability and observability of path and cycle graphs”, IEEE Trans. Autom. Control, vol. 57 (2012), 743–748.
J. F. Ritt, “Prime and composite polynomials”, Trans. Am. Math. Soc., vol. 23(1), (1922), 51–66.
H.H. Rosenbrock, State-Space and Multivariable Theory, Wiley, New York, 1970.
H.H. Rosenbrock and A.C. Pugh, “Contributions to a hierarchical theory of systems”, Int. J. Control, vol. 19, (1974), 845–867.
M.C. Smith, “Synthesis of mechanical networks: the inerter”, IEEE Trans. Autom. Contr. vol. 47, (2002), 1648–1662.
E.D. Sontag, “On finitary linear systems”, Kybernetika, vol. 15, (1979), 349–358.
S. Sundaram and C.N. Hadjicostis, “Control and estimation in finite state multi-agent systems: A finite field approach”, IEEE Trans. Autom. Control, vol. 58, (2013), 60–73.
H.G. Tanner, “On the controllability of nearest neighbor interconnections”, Proc. 43rd IEEE Conference on Decision and Control, Atlantis, Bahamas (2004), 2467–2472.
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Fuhrmann, P.A., Helmke, U. (2015). Interconnected Systems. In: The Mathematics of Networks of Linear Systems. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-16646-9_9
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DOI: https://doi.org/10.1007/978-3-319-16646-9_9
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