Minimum sensitivity realizations of networks of linear systems

Uwe  Helmke; Michael  Schönlein

doi:10.3934/naco.2016010

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

Minimum sensitivity realizations of networks of linear systems

Uwe Helmke ^1, and Michael Schönlein ^2,

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

Abstract
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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.

Keywords: linear systems, sensitivity, balancing, optimization., Networks.

Mathematics Subject Classification: Primary: 93B35; Secondary: 93C0.

Citation: Uwe Helmke, Michael Schönlein. Minimum sensitivity realizations of networks of linear systems. Numerical Algebra, Control & 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. Google Scholar
[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. Google Scholar
[3]	R. Fornaro, Numerical evaluation of integrals around simple closed curves, SIAM J. Numer. Anal., 10 (1973), 623-634. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer, London, 1994. doi: 10.1007/978-1-4471-3467-1. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	J. Lunze, Control Theory of Digitally Networked Dynamic Systems, Cham: Springer, 2014. doi: 10.1007/978-3-319-01131-8. Google Scholar
[16]	M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010. doi: 10.1515/9781400835355. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	G. Zames, Functional analysis applied to nonlinear feedback systems, IEEE T. Circuits Syst., 10 (1963), 392-404. Google Scholar

show all references

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. Google Scholar
[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. Google Scholar
[3]	R. Fornaro, Numerical evaluation of integrals around simple closed curves, SIAM J. Numer. Anal., 10 (1973), 623-634. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer, London, 1994. doi: 10.1007/978-1-4471-3467-1. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	J. Lunze, Control Theory of Digitally Networked Dynamic Systems, Cham: Springer, 2014. doi: 10.1007/978-3-319-01131-8. Google Scholar
[16]	M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010. doi: 10.1515/9781400835355. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	G. Zames, Functional analysis applied to nonlinear feedback systems, IEEE T. Circuits Syst., 10 (1963), 392-404. Google Scholar

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