American Institute of Mathematical Sciences

2016, 6(3): 241-262. doi: 10.3934/naco.2016010

Minimum sensitivity realizations of networks of linear systems

 1 Institute for Mathematics, University of Würzburg, Emil-Fischer Straße 40, 97074 Würzburg, Germany 2 Institute for Mathematics, University of Würzburg, Emil-Fischer Straße 40, 97074 Würzburg, Germany

Received  April 2015 Revised  July 2016 Published  September 2016

We investigate networks of linear control systems that are interconnected 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 Gramians that determine minimum sensitivity state space realizations of the network. Our work extends previous work by [6,10,11] on $L^2$-minimum sensitivity design.
Citation: Uwe Helmke, Michael Schönlein. Minimum sensitivity realizations of networks of linear systems. Numerical Algebra, Control and Optimization, 2016, 6 (3) : 241-262. doi: 10.3934/naco.2016010
References:
 [1] J. B. Cruz and W. R. Perkins, A new approach to the sensitivity problem in multivariable feedback system design, IEEE T. Automat. Contr., 9 (1964), 216-223. [2] D. F. Delchamps, New geometric approaches to parameter sensitivity in feedback systems, in Modelling, Identification and Robust Control(eds. C. I. Byrnes and A. Lindquist), Elsevier Science Publishers B. V. (North-Holland), (1986), 445-456. [3] R. Fornaro, Numerical evaluation of integrals around simple closed curves, SIAM J. Numer. Anal., 10 (1973), 623-634. [4] P. A. Fuhrmann and U. Helmke, Reachability, observability and strict equivalence of networks of linear systems, Math. Control Signal., 25 (2013), 437-471. doi: 10.1007/s00498-012-0104-0. [5] P. A. Fuhrmann and U. Helmke, The Mathematics of Networks of Linear Systems, Cham: Springer, 2015. doi: 10.1007/978-3-319-16646-9. [6] M. Gevers and G. Li, Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects, Springer, Berlin, 1993. doi: 10.1007/978-1-4471-2039-1. [7] S. Hara, A unified approach to decentralized cooperative control for large-scale networked dynamical systems, in Perspectives in Mathematical System Theory, Control, and Signal Processing (eds. J. C. Willems et al.), Springer, Berlin, (2010), 61-72. doi: 10.1007/978-3-540-93918-4_6. [8] S. Hara, T. Hayakawa and H. Sugata, LTI systems with generalized frequency variables: A unified framework for homogeneous multi-agent dynamical systems, SICE Journal of Control, Measurement, and Systems Integration, 2 (2009), 299-306. [9] U. Helmke, I. Kurniawan, P. Lang and M. Schönlein, Sensitivity optimal design of networks of identical linear systems, in Proc. Mathematical Theory of Networks and Systems (MTNS2012), Melbourne, Australia, 5-9 July 2012, paper 0283. [10] U. Helmke and J. B. Moore, L2 sensitivity minimization of linear system representations via gradient flows, J. Math. Syst. Estim. Control, 5 (1995), 79-98. [11] U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer, London, 1994. doi: 10.1007/978-1-4471-3467-1. [12] R. A. Horn and R. Mathias, Block-matrix generalizations of Schur's basic theorems on Hadamard products, Linear Algebra Appl., 172 (1992), 337-346. doi: 10.1016/0024-3795(92)90033-7. [13] S. Koshita, M. Abe, and M. Kawamata, Analysis of second-order modes of linear discrete-time systems under bounded-real transformations, IEICE T. Fund. Electr., 90 (2007), 2510-2515. [14] S. Liu, Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra Appl., 289 (1998), 267-277. doi: 10.1016/S0024-3795(98)10209-4. [15] J. Lunze, Control Theory of Digitally Networked Dynamic Systems, Cham: Springer, 2014. doi: 10.1007/978-3-319-01131-8. [16] M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010. doi: 10.1515/9781400835355. [17] B. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE T. Automat. Contr., 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568. [18] C. T. Mullis and R. A. Roberts, Roundoff noise in digital filters: frequency transformations and invariants, IEEE T. Acoust. Speech, 24 (1976), 538-550. [19] C. T. Mullis and R. A. Roberts, Synthesis of minimum roundoff noise fixed point digital filters, IEEE T. Circuits Syst., 23 (1976), 551-562. [20] R. Olfati-Saber, J. Fax, and R. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233. [21] L. Pernebo and L. M. Silverman, Model reduction via balanced state space representations, IEEE T. Automat. Contr., 27 (1982), 382-387. doi: 10.1109/TAC.1982.1102945. [22] V. Tavsanoglu and L. Thiele, Optimal design of state-space digital filters by simultaneous minimization of sensitivity and roundoff noise, IEEE T. Automat. Contr., 31 (1984), 884-888. doi: 10.1109/TCS.1984.1085426. [23] L. Thiele, On the sensitivity of linear state-space systems, IEEE T. Circuits Syst., 33 (1986), 502-510. doi: 10.1109/TCS.1986.1085951. [24] W.-Y. Yan, J. B. Moore and U. Helmke, Recursive algorithms for solving a class of nonlinear matrix equations with applications to certain sensitivity optimization problems, SIAM J. Control Optim., 32 (1994), 1559-1576. doi: 10.1137/S0363012992226855. [25] G. Zames, Functional analysis applied to nonlinear feedback systems, IEEE T. Circuits Syst., 10 (1963), 392-404.

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References:
 [1] J. B. Cruz and W. R. Perkins, A new approach to the sensitivity problem in multivariable feedback system design, IEEE T. Automat. Contr., 9 (1964), 216-223. [2] D. F. Delchamps, New geometric approaches to parameter sensitivity in feedback systems, in Modelling, Identification and Robust Control(eds. C. I. Byrnes and A. Lindquist), Elsevier Science Publishers B. V. (North-Holland), (1986), 445-456. [3] R. Fornaro, Numerical evaluation of integrals around simple closed curves, SIAM J. Numer. Anal., 10 (1973), 623-634. [4] P. A. Fuhrmann and U. Helmke, Reachability, observability and strict equivalence of networks of linear systems, Math. Control Signal., 25 (2013), 437-471. doi: 10.1007/s00498-012-0104-0. [5] P. A. Fuhrmann and U. Helmke, The Mathematics of Networks of Linear Systems, Cham: Springer, 2015. doi: 10.1007/978-3-319-16646-9. [6] M. Gevers and G. Li, Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects, Springer, Berlin, 1993. doi: 10.1007/978-1-4471-2039-1. [7] S. Hara, A unified approach to decentralized cooperative control for large-scale networked dynamical systems, in Perspectives in Mathematical System Theory, Control, and Signal Processing (eds. J. C. Willems et al.), Springer, Berlin, (2010), 61-72. doi: 10.1007/978-3-540-93918-4_6. [8] S. Hara, T. Hayakawa and H. Sugata, LTI systems with generalized frequency variables: A unified framework for homogeneous multi-agent dynamical systems, SICE Journal of Control, Measurement, and Systems Integration, 2 (2009), 299-306. [9] U. Helmke, I. Kurniawan, P. Lang and M. Schönlein, Sensitivity optimal design of networks of identical linear systems, in Proc. Mathematical Theory of Networks and Systems (MTNS2012), Melbourne, Australia, 5-9 July 2012, paper 0283. [10] U. Helmke and J. B. Moore, L2 sensitivity minimization of linear system representations via gradient flows, J. Math. Syst. Estim. Control, 5 (1995), 79-98. [11] U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer, London, 1994. doi: 10.1007/978-1-4471-3467-1. [12] R. A. Horn and R. Mathias, Block-matrix generalizations of Schur's basic theorems on Hadamard products, Linear Algebra Appl., 172 (1992), 337-346. doi: 10.1016/0024-3795(92)90033-7. [13] S. Koshita, M. Abe, and M. Kawamata, Analysis of second-order modes of linear discrete-time systems under bounded-real transformations, IEICE T. Fund. Electr., 90 (2007), 2510-2515. [14] S. Liu, Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra Appl., 289 (1998), 267-277. doi: 10.1016/S0024-3795(98)10209-4. [15] J. Lunze, Control Theory of Digitally Networked Dynamic Systems, Cham: Springer, 2014. doi: 10.1007/978-3-319-01131-8. [16] M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010. doi: 10.1515/9781400835355. [17] B. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE T. Automat. Contr., 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568. [18] C. T. Mullis and R. A. Roberts, Roundoff noise in digital filters: frequency transformations and invariants, IEEE T. Acoust. Speech, 24 (1976), 538-550. [19] C. T. Mullis and R. A. Roberts, Synthesis of minimum roundoff noise fixed point digital filters, IEEE T. Circuits Syst., 23 (1976), 551-562. [20] R. Olfati-Saber, J. Fax, and R. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233. [21] L. Pernebo and L. M. Silverman, Model reduction via balanced state space representations, IEEE T. Automat. Contr., 27 (1982), 382-387. doi: 10.1109/TAC.1982.1102945. [22] V. Tavsanoglu and L. Thiele, Optimal design of state-space digital filters by simultaneous minimization of sensitivity and roundoff noise, IEEE T. Automat. Contr., 31 (1984), 884-888. doi: 10.1109/TCS.1984.1085426. [23] L. Thiele, On the sensitivity of linear state-space systems, IEEE T. Circuits Syst., 33 (1986), 502-510. doi: 10.1109/TCS.1986.1085951. [24] W.-Y. Yan, J. B. Moore and U. Helmke, Recursive algorithms for solving a class of nonlinear matrix equations with applications to certain sensitivity optimization problems, SIAM J. Control Optim., 32 (1994), 1559-1576. doi: 10.1137/S0363012992226855. [25] G. Zames, Functional analysis applied to nonlinear feedback systems, IEEE T. Circuits Syst., 10 (1963), 392-404.
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